Solving Mathematics Practice

Introduction to solving mathematics practice:

In this section we will see about solving mathematics practice. It includes solving all types of mathematics practice problems with neat and clear explanation along with each and every step of solution. Basically mathematics includes algebra, geometry, number system, trigonometry, graphs and maps, Probability and statistics, measurement of time, and so on. Let us see about solving mathematics practice.

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Example problems for solving mathematics practice:


Example problem 1: Write the expression for the following:” g is added by 12 and decreased by 5”

Solution:

Given g is added by 12 and decreased by 5

g + 12 -5 = g + 7

Answer: g + 7

Example problem 2: Mrs. Michael baked apple pies for her family. The boys ate `3/5 ` of a pie and the girls ate `1/4` of a pie. How much more pie did the boys eat than the girls?

Solution:

We have to find how much more pie the boys ate by subtracting `1/4` from `3/5` .

You can use 5 as the common denominator.

Multiply the numerator and denominator of `1/4` by 5:

`1/4 * 5/5 = 5/20`

Multiply the numerator and denominator of `3/4 ` by 4:

` 3/4 * 4/4 = 12/20`

Now subtract: `12/20 - 5/20 = 7/20`

Answer: The boys ate `7/20` of a pie more than the girls ate.


Practice problems for solving mathematics practice:


Practice problem 1: In the following 85, 355 find the number is in the thousands place?

Practice problem 2: Vikki jogged `1/2 ` of a lap in playing class and `3/4 ` of a lap during track practice. How many laps did Vikki jog in all?

Practice problem 3: James has $150 in a savings account that earns 3% interest per year. How much interest will he earn in one year?

Practice problem 4: Katrina decided to take the train from Malaysia to Singapore. He got on the train in Malaysia at half past eight. The train took nine hours and fifty minutes to get to Singapore. What time was it when Katrina got off the train?

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Solutions for solving mathematics practice:


Solution 1: 5,000

Solution 2: Vikki jogged `1 1/4 ` lap in all.

Solution 3: James earned $4.5 in one year.

Solution 4: Katrina got off the train at 6:40.

Study Basic Mathematics Test

Introduction to study basic mathematics test:

Let us see about the topic is basic math tests help. In mathematics subject have special problems with a mixture of methods. These are getting practiced using test method for our personality estimate. In basic math tests help means addition, subtraction, multiplication and division using different types we can tested design. Given some test problems and with solution these based on the topic is basic math tests help.

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Sample questions for using study basic mathematics test:

Test question 1: Find the Greatest common factors of 16 and 24.

Test question 2: Is 983 a prime number or composite number?

Test question 3: Multiply and simplify the following: 2.365 * 15 (give your answer in decimal fraction format)

Test question 4: Louie buys 10 packets of color pencils for $85. The envelopes all have the equal cost. Find the price of each packet.

Test question 5: Find the area of the rectangle. The rectangle’s length and breadth is 22 cm and 8 cm respectively.

Test question 6: Multiply the following terms: 2500 * 800

Test question 7: Evaluate: 5!

Test question 8: Evaluate the following given expression: 86 + m, if m = 4

Test question 9: Find the difference: 31.562 – 1.0003

Test question 10: Find the perimeter of the regular hexagon, whose side is 4 cm.

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Solutions for using study basic mathematics test:


Answer key 1: The Greatest common factors of 16 and 24 is 8.

Answer key 2: 983 is a prime number. Since the given digit is divisible by one and itself only.

Answer key 3: 35.475

Answer key 4: The price of each packet is $8.5

Answer key 5: The area of the rectangle is 176 sq cm.

Answer key 6: 2000000

Answer key 7: 120

Answer key 8: 90

Answer key 9: 30.5617

Answer key 10: The perimeter of the regular hexagon is 24 cm.

In this section we have solved about study basic mathematics test.

Pure Mathematics Degree

Introduction to pure mathematics degree:

Pure Mathematics is planned for students with a strong mathematical environment and also helpful for higher graduate students.  The subject mathematics contains calculus, differential functions, unit measurements, number sense, groups, numerical methods, fractions & mixed numbers, vectors, algebra, geometry, algebra function, probability and statistics number using words decimals. In this article we shall discuss about pure mathematics degree. I like to share this Vector Calculus Identities with you all through my article.


Problem on differential function- Pure mathematics degree:


Example problem1:

To find `f` `^'(x)` the function of` f(x) = x^2+2x+3,` when `x=2.`

Solution:

`f(x) = x^2+2x+3`

`f^'(x)=2x+2`

`f^'(2) = 2(2)+2`

`f^'(2) = 4+2`

`f^'(2) =6`

Answer is` 6.`

Example problem2:

To find `f^'(x) ` the function of` f(x) = x^2+2x+3,` when `x=3.`

Solution:

`f(x) = x^2+2x+3`

`f^'(x)=2x+2`

`f^'(2) = 2(3)+2`

`f^'(2) = 6+2`

`f^'(2) =8`

Answer is `8.`

Example problem3:

To find` f^'(x)` the function of `f(x) = x^2+2x+` `3,` when `x=4.`

Solution:

`f(x) = x^2+2x+3`

`f^'(x)=2x+2`

`f^'(2) = 2(4)+2`

`f^'(2) = 8+2`

`f^'(2) =10`

Answer is `10.`

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Calculus problems-Pure mathematics degree:


problem 1:- Pure mathematics degree

Integrate the known expression with respect to `x: int 12x^4 - 11x^5 dx`

Solution:

Given` int 12x^4 - 11x^5 dx.`

Step 1:-

`int 12x^4- 11x^5 dx = int 12x^4 dx. - int 11x^5 dx.`

Step 2:-

`= int 12x^4dx. - 11 int x^5 dx.`

Step 3:-

`= (12x^5)/ (5) - (11x^6)/ (6) + c.`

Step 4:-

`int 12x^4 - 11x^5 dx = (12x^5)/ (5) - (11x^6)/ (6) + c.`

Answer:

`int 12x^4 - 11x^5 dx = (12x^5)/ (5) - (11x^6)/ (6) + c`

problem 2:- Pure mathematics degree

Integrate the known exponential function: ` int tan x + e^ (2x) dx`

Solution:

Step 1:-

`int tan x + e^ (2x) dx = int tan x dx + int (e^(2x)) dx`

`= int tan x dx + e^ (2x)/ (2)`

Step 2:-

`= - log (cos x) + e^ (2x)/ (2) + c`

Answer:

`- log (cos x) + e^ (2x)/ (2) + c`

Range Mean in Mathematics

Introduction on what does range mean in mathematics:

The length of the interval which is small by containing all data is defined as the range. In mathematics it is used in the identification for the dispersion in statistics. The measurement of the range is based on the poor as well as weak measure of the data. The range is measured by using the same units. For example, if range is used in measuring population it uses some of the mean distribution. I like to share this Inter Quartile Range with you all through my article.


What does range mean in mathematics:-Range definition


Definition of range which is used in mathematics:

Range is the simplest measure of dispersion.

It is defined as the difference between the biggest and the smallest values in the series. The equation for range defining what it means in mathematics is given as,

Range = Vmax - Vmin

where Vmax is the maximum value,

Vmin is the minimum value.

Coefficient of range:

In mathematics the coefficient of range is calculated by using the formula which means the range is given as,

Coefficient of range = `(Vmax - Vmin)/(Vmax + Vmin)`

Median in mathematics:

The median is used in the mathematics which means what is the middle term from the given data. The middle term in mathematics is found by arranging the data in ascending and descending order.

If the number of data is in odd number then the middle term is considered as the median which means the middle term. If the number of data is in even number then the middle two terms are added and divided by two for getting the median.

Mode in mathematics:

The mode is the term which means identifying the frequently occurred terms.

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Example problems to learn what does the range mean in mathematics


1. The datas are 15, 30, 40, 45, 56, 64. What is the range for the given data?

The maximum value = 64

Minimum value = 15

Range = Vmax - Vmin

= 64 - 15

= 49

Coefficient of range = `(Vmax - Vmin)/(Vmax + Vmin)`

= `(64 - 15)/(64 + 15)`

=   `(49)/(79)`

= 0.625

2. The largest value of the data is 72. If the range of data is 25, what is the smallest value of the data?

Maximum value = 72

Range = 25

Minimum value = ?

Range = Vmax - Vmin

25 = 72 - Vmin

Vmin = 72 - 25

= 47

Hence 47 is the minimum value.

3. What is the mode of the following numbers 11, 12, 6, 5, 6, 8, 6?

Here the mode is 6 because it occurs three times in the given numbers.

4. What is the mode of the given data 7, 11, 4, 11, 13, 14, 4, 4, 11?

Here the mode is 11 and 4 since both the numbers occur three times.

5. Find the median for the following numbers 2, 4, 6, 8, 7, 10, 9?

Step 1: Arrange the numbers in ascending order.

2, 4, 6, 7, 8, 9, 10.

Step 2: Since the given numbers are odd then the middle term is 7.

6. What is the median for the following numbers 3, 6, 7, 4, 2, 11.

Step 1: Arrange the numbers in ascending order.

2, 3, 4, 6, 7, 11.

Step 2: Since the given numbers are even then add teh two middle terms.

Median = `(4+6)/(2)`

= `(10)/(2)`

= 5.

Hence the median is 5.

Learn Basic Mathematics Online Study

Introduction to study for learn online basic mathematics:
Mathematics is used in many fields like, engineering, science, and medicine. In basic math the students can receive first level of math education called primary education. Online learning the session available 24x7 so students can study and get help any time. In this article we shall learn to study online basic mathematics example problems. Understanding Constructing Parallel Lines is always challenging for me but thanks to all math help websites to help me out.


Learn online basic mathematics study example problem


Example:

Find 10xy – 18xy

Solution:

Here the subtraction is possible because the given terms are like terms, but here we have to subtract the grater number from a small number.

10y – 18xy = (10 – 18)xy

= – 8xy

Example:

Solve for x:

8 x - 2 = 4 x + 14

Solution:

8 x - 2 = 4 x + 14

Subtract 4x from both sides of the equation

8x – 4x – 2 = 4x – 4x + 14

4x – 2 = 14

Add 2 to both sides of the equation

4x – 2 + 2 = 14 + 2

4x = 16

Divided by 4 both side of the equation

x = 4

Example:

Find the ratio of 40 cm to 8meter.

Solution:

8meter = 8 × 100cm = 800 cm

Therefore required ratio = 40: 800

= 4: 80

= 1: 20

Example:

A room in rectangle form the length of the room is 20cm and the breadth of the room is 15cm. Find the room area and perimeter.

Solution:

Room length is l= 20cm
Room breadth, b= 15cm
Area of rectangle formula A = l x b sq units

= 20 x 15

= 300

Therefore the room area is A = 300 sq.cm

Perimeter of rectangle room, P        = 2l + 2b units

= 2 x 20 + 2 x 15

= 40 + 30

= 70

Therefore perimeter of the room is, P = 70 cm

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Learn online basic mathematics study example problem

Problem:

Find 10xy – 15xy

Answer: – 5xy

Problem:

A room in rectangle form the length of the room is 15cm and the breadth of the room is 10cm. Find the room area and perimeter.

Answer:

The room area is A = 150 sq.cm

Perimeter of the room = 50 cm

Recreational Mathematics

Recreational mathematics is an umbrella term, referring to mathematical puzzles and mathematical games. Not all problems in this field require knowledge of advanced mathematics, and thus, recreational mathematics often attracts the curiosity of non-mathematicians, and inspires their further study of mathematics. Please express your views of this topic Business Mathematics by commenting on blog.


What is recreational mathematics all about?


Recreation mathematics is all about fun games, puzzles, brain teasers, number games etc. This recreational mathematics is genre of mathematics includes logic puzzles and other puzzles that require deductive reasoning, the aesthetics of mathematics, and peculiar or amusing stories and coincidences about mathematics and mathematicians. Some of the more well-known topics in recreational mathematics are magic squares and fractals.

Recreational mathematics is very much interesting topic for motivating the children who lack in grasping the basics in mathematics. In general the recreational mathematics is used as a hobby among the kids to make them learn while playing some games and puzzles. Recreation mathematics allows the weak students to pick up mathematics in no time. This mathematic games and puzzles are recommended be all schools and study centers to promote the interest of a child.

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Where all recreational mathematics is used;


The recreational games are not necessarily involves any mathematical process of adding or subtracting any numbers, it’s just a fun game to make them understand the fundamentals and basics. All this games carries some rules and regulations to follow but not much of calculations involved. There is a chance of interaction between the players which widens the horizon of learning some techniques without any risk. The recreational mathematic is advised at the early stage of the schooling amongst the growing children.

Lots of short stories which mentions about this recreational mathematics are seen in libraries which can be recommended. Normal way playing can have this type of mathematical problems under which any child can grasp the basics of this subject. Recreational mathematics should also be played during leisurely time of family members or during holidays with friends and relatives. At the end we conclude with the opinion of having compulsory mathematical games in daily routine will help in improving the skill of a child.

Learn Mathematics Solutions

Introduction to learn mathematics solutions:

In mathematics solutions, we can learn many topics like algebra, differential calculus, trigonometry, geometry, etc. Here we can see mathematics functions and mathematics equation in algebra.Mathematics functions are help to learn one or more degree polynomial function with one or more variables. The example to learn mathematics function is, f(x) = 5x + 11, `f(x) = x^2 + 8x - 5`

Mathematics equations are help to learn one or two variables with one or more order values. The example to learn mathematics equation is y = 15x + 5. `y = 5x^2 + 8x - 2`


Examples to learn mathematics equation solutions:


Learn solutions for mathematics equations example problem 1:

Learn the factors for the given mathematics equations,` x^2 - 5x + 4 = y`

Solutions:

Plug y = 0, to find the factor for the given mathematics equation.

We can separate the equation as sum and product of roots,

This is in the form of

`x^2` + (Sum of the roots) x + (Product of the roots) = 0

By comparing the given equation we can get,

Sum of the roots = -5

Product of the roots = 4

The possible number of outcome for product of the roots is `(-4)xx (-1) and (4)xx(1)`

By comparing the product of root value, we can obtain the sum of the roots,

To obtain the value for sum of the roots -5, consider (-4) + (-1)

Now substitute this sum of roots values in the equation, we get

`x^2 - 4x - x + 4 = 0`

Now, bring out x as common in the first two terms and -1 as common in the next two terms

x(x - 4) - 1 (x - 4) = 0

(x - 4) (x- 1) = 0

Thus, the factors for the given mathematics equation is (x - 4) (x - 1)

Learn solutions for mathematics equations example problem 2:

Learn the given mathematics equation by reducing the given expression, 3(5y + 8) = 8 – 5x.

Solutions:

Step 1: Given expression,

3(5y + 8) = 8 - 5x

Step 2: By the use of distribute property

15y + 24 = 8 - 5x

Step 3: Add -24 on either side,

15y + 24 – 24 = 8 - 5x - 24

15y = -16 – 5x

Step 4: Divide by 15 on either side

` (15y) / (15) = (-16)/ (15) - (5x)/ (15)`

Step 5: By simplifying, the expression we get,

` y = (-16)/ (15) - (x)/ (3).`

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Examples to learn mathematics functions solutions:


Learn solutions for mathematics functions example problem 1:

By reducing the expression 8x - 5 = 8y - 12. Find the mathematics function.

Solutions:

Step 1: Given expression

8x - 5 = 8y - 12

Step 2: Add 12 on either side, we get

8x - 5 + 12 = 8y - 12 + 12

8x + 7 = 8y

Step 3: Divide by 8 on either side, we get

`(8x) /8 + 7/8 = (8y) /8`

Step 4: By simplifying the above expression, we get

` x + 7/8 = y`

Step 5: Replace y = f(x), we get

`x + 7/8 = f(x)`

Thus, we obtain the mathematics function is `f(x) = x + 7/8.`

Learn Mathematics Definitions

Introduction to learn mathematics definitions:
Mathematics is a vast area, in which plenty of definitions available to learn under various branches. Learning mathematics definitions is very important because it helps to understand various concepts involved in maths and also helps to solve various problems involved in math. In this article learn mathematics definitions, we are going to learn few definitions involved in math.


Mathematics definitions


Integers: All the positive and negative whole numbers are referred as integers.

Parallel lines: If the two distinct coplanar lines do not intersect, they are called coplanar points.

Postulate: Postulate is nothing but a statement that is accepted as true without proof.

Quadrilateral: Quadrilateral is nothing but a polygon with four sides.

Central angle: It is the angle present in the circle, in which the vertex is located at the center of the circle.

Ray: Ray is nothing but a part of the line, which starts at a particular point and extends infinitely in one direction.

x-intercept: If a point at which the graph intersects the x-axis, it is referred as x - intercept.

Straight angle: Angle whose measure 180 degrees is known as straight angle.

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Additional mathematics definitions:


Number line: Number line is nothing but a line that represents the set of all real numbers.

Outlier: Outlier is a data point, which is particularly separate from the rest of all other data points.

Chord: Chord is nothing but the line segment located on the interior of the circle.

Volume: Volume is nothing but the total amount of space, which is enclosed in a solid.

y-intercept: If a point at which the graph intersects the y-axis, it is referred as y - intercept.

Diameter: Diameter is nothing but a line segment between two points on the circle or sphere that passes through the center.

z-intercept: If a point at which the graph intersects the z-axis, it is referred as z - intercept.

Solving Number Theory Problems

Introduction of solving number theory problems
Number theory is the theory about numbers which is called "the queen of mathematics" by the legendary mathematician Carl Friedrich Gauss, number theory is one of the oldest and largest branches of pure mathematics. The number theory delves deep into the structure and nature of numbers, and explores the remarkable, often beautiful relationships among them.

The number theory has many different types of numbers:

Solving Natural numbers problemssolving Prime numbers problems
solving Integers problems
solving Algebraic numbers problems
solving imaginary numbers problems
solving transcendental numbers problems


Example problems for solving number theory


Example: 1

Find the values of consecutive numbers where the sum of the two numbers is 161.

Solution:

Let as assume the two consecutive numbers be x, x+1.

Where the sum is 161 so,

x + x + 1 = 161

2x + 1 = 161

2x = 160

x = 80

Therefore, x + 1 = 80 + 1 = 81

So the consecutive numbers is 80 and 81.

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Example: 2

Find all integers n such that n − 18 and n + 18 are both perfect Squares.

Solution:            Let as assume n −18 = a2 and n +18 = b2.

Then b2 −a2 = 36, so (b−a) (b+ a) = 22 .32.

Because b−a and b+a are of the same parity,

So, the following possibilities are:  b − a = 2, b + a = 18, yielding b = 10, a = 8,

And b − a = 6, b+a = 6, yielding a = 0, b = 6.

Hence the integers with this property are n = 46 and n = 18.


Example :3


Find the values of consecutive numbers where the product of the two numbers is 121.

Solution:-

Let as assume the two consecutive numbers be x, x +1.

Where the product is 121 so,

x * (x + 1) = 121

x^2 + x = 121

x^2 + x – 121 = 0

x^2 + 12x – 11x – 121 = 0

(x + 12) (x - 11) = 0

(x + 12) = 0 (or) (x - 11) = 0

x = -12 (or) x = 11

-12 is not possible to get 121

Therefore we take x = 11

So, x + 1 = 11 + 1 = 12

The consecutive numbers is 11 and 12.

Mathematics Tangrams

Introduction to mathematics tangrams:

Among puzzles Tangram is certainly the mainly outstanding of each one. Tangram originates through China. Not everything is standard regarding its inventor or else correctly while the puzzle is invented. Eliminate it is recognized to be particularly accepted in China as of on 1800. The initial existing Chinese reserve lying on tangrams is obtainable in 1813. Having problem with Angle Obtuse keep reading my upcoming posts, i will try to help you.


Mathematics tangrams:

Some say in mathematics tangrams are typically played on residence during women as well as kids. This existence, the tangram is a problem game to know how to be enjoyed through the whole relations. It does not need an excessive quantity of ability. It now requires persistence, instance with mind. Someone by a part of article also a position of scissors know how to include consider of tangram, although particularly respected tangram locate include be entire since of carefully engraved ivory, tortoise casing with mother-of-pearl.

The confront of mathematics tangram be to organize seven uncomplicated geometrical part recognized tans two huge triangles, two undersized triangles with a rectangle within all type of behavior to build shape to symbolize community, substance also you know how to consider of. Our commission is just to reconstruct the shape, with every the seven tans exclusive of be related. But, several qualified puzzle solvers declare to group typically overrate their capability the complexity of the problem on the establishment period of live.

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Analysis tangrams:

According to statement, people who utilize their left understanding further lean to center lying on logical thoughts with accurateness. Those who utilize the right brain further center lying on aesthetics, reaction as well as originality.

Mathematics tangram knows how to further a further entire rational with aesthetic understanding. Skilled illustration - spatial thinker might locate to resolve mathematics tangrams exercise their valid analysis capability. And systematic academic might locate it improve their ability in concert by form, color along with thoughts.

The tangram is an analysis puzzle consisting of seven plane form, identified tans that are place mutually toward form shapes. The purpose of the problem is to structure a definite form by all seven parts that might not be related.

Two Perfect Squares

Definition:

In mathematics, a square number, sometimes also called a perfect square, is an integer that is the square of an integer; in other words, it is the product of some integer with itself. So, for example, 9 is a square number, since it can be written as 3 × 3. Square numbers are non-negative. Another way of saying that a (non-negative) number is a square number is that its square root is again an integer. Having problem with Solid Geometry keep reading my upcoming posts, i will try to help you.


Examples


For example 1,

(x + 2)(x + 2)

You get:

x^2 + 4x + 4

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Therefore, the quadratic expression x^2 + 4x + 4 is a perfect square since it factors into two identical binomials which are (x + 2) and (x + 2).

Notice that (x + 2) (x + 2) can be written (x + 2)2. So:

x^2 + 4x + 4 = (x + 2)2

For example 2,

(x + 3)(x + 3)

You get:

x^2 + 6x + 9

Therefore, the quadratic expression x^2 + 6x + 9 is a perfect square since it factors into two identical binomials which are (x + 3) and (x + 3).

Notice that (x + 3) (x + 3) can be written (x + 3)2. So:

x^2 + 6x + 9 = (x + 3)2

For example 3,

(x + 4)(x + 4)

You get:

x^2 + 8x + 16

Therefore, the quadratic expression x^2 + 8x + 16 is a perfect square since it factors into two identical binomials which are (x + 4) and (x + 4).

Notice that (x + 4) (x + 4) can be written (x + 4)2. So:

x^2 + 8x + 16 = (x + 4)2

For example 4,

(x + 5)(x + 5)

You get:

x^2 + 10x + 25

Therefore, the quadratic expression x^2 + 10x + 25 is a perfect square since it factors into two identical binomials which are (x + 5) and (x + 5).

Notice that (x + 5) (x + 5) can be written (x + 5)2. So:

x^2 + 10x + 25 = (x + 5)2

For example 5,

(x + 6)(x + 6)

You get:

x^2 + 12x + 30

Therefore, the quadratic expression x^2 + 12x + 30 is a perfect square since it factors into two identical binomials which are (x +6) and (x + 6).

Notice that (x + 6) (x + 6) can be written (x + 6)2. So:

x^2 + 12x + 30 = (x + 6)2

For example 6,

(x -5)(x -5)

You get:

x^2 - 10x + 25

Therefore, the quadratic expression x^2 - 10x + 25 is a perfect square since it factors into two identical binomials which are (x - 5) and (x - 5).

Notice that (x - 5) (x - 5) can be written (x - 5)2. So:

x^2 - 10x + 25 = (x - 5)2

Discrete Mathematics Notes

Introduction to discrete mathematics notes:

Discrete Mathematics deals with several selected topics in Mathematics that are essential to the study of many Computer Science areas. Since it is very difficult to cover all the topics, only two topics, namely “Mathematical Logic” and “Groups” have been introduced. These notes will be very much helpful to the students in certain practical applications related to Computer Science. In this article we shall discuss about discrete mathematics notes.

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Discrete mathematics notes:


Logical statement or Proposition:

A statement or a proposition is a sentence which is either true or false but not both.

A sentence which is both true and false simultaneously is not a statement, rather it is a paradox.

Example 1:

(a) Consider the following sentences:

(i) The earth is a planet.

(ii) Rose is a flower.

Truth value of a statement:

The truth of a statement is called its truth value. If a statement is true, we say that its truth value is TRUE or T and if it is false, we say that its truth value is FALSE or F.

Simple statements:

A statement is said to be simple if it cannot be broken into two or more statements. All the statements in (a) and (b) of Example 1 are simple statements.

Compound statements:

If a statement is the combination of two or more simple statements, then it is said to be a compound statement.

Conjunction:

If two simple statements p and q are connected by the word ‘and’, then the resulting compound statement ‘p and q’ is called the conjunction of p and q and is written in the symbolic form as ‘p ? q’.

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Discrete mathematics notes problems:

Example 1:

(i) Show that ((~ p) ? (~ q)) ? p is a tautology.

Solution:

(i) Truth table for ((~ p) ? (~ q)) ? p

p          q         ~ p      ~ q      (~ p) ? (~ q)     ((~ p) ? (~ q))? p

T          T          F            F                      F                      T

T          F          F           T                      T                      T

F          T          T          F                      T                      T

F          F          T          T                      T                      T

The last column contains only T. Therefore ((~ p) ? (~ q)) ? p is a tautology.

Example 2:

Show that ((~ q) ? p) ? q is a contradiction.

Truth table for ((~ q) ? p) ? q

p          q          ~ q      (~ q) ? p          ((~ q) ? p) ? q

T          T          F          F                                  F

T          F          T          T                                  F

F          T          F          F                                  F

F          F          T          F                                  F

The last column contains only F. ? ((~ q) ? p) ? q is a contradiction.

Discrete Mathematics Sample

Introduction discrete mathematics sample

Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic do not vary smoothly in this way, but have distinct, separated values Discrete mathematics therefore excludes topics in "continuous mathematics" such as calculus and analysis. Having problem with Graph Ordered Pairs keep reading my upcoming posts, i will try to help you.

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Discrete mathematics sample explanations:


Here will study about the discrete mathematics problems.

The discrete mathematics contains the set of topics. These topics are cover in Tautologies and Logical Equivalence

Sentential Functions and Sets are in logic and sets. Relation and functions,   Equivalence Relations Equivalence Classes. Natural numbers, division and factorization.  Division  ,Factorization ,

Greatest Common Divisor. these are the some of the discrete mathematics

Here we will see some of the samples in the discrete mathematics

Example for sentence

1.The sentence \1 + 2 = 3 and 2 + 2 = 4" is true.

2.The sentence \3 + 3 = 6 and _ is rational" is false.

Example for relations and functions

Definition. Let A and B be sets. By a relative R on A and B, we mean a subset of the Cartesian

Product a x b.

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Discrete mathematics sample problems:


Here we will learn about the discrete mathematics sample problems

The natural numbers

Example1:

The set of natural numbers is usually given by

N = {1, 2, 3 …}

Division

Examples

Division we divide 24 by 4

Solution:

24/4 =6

We divide 24 by 4 we get answer is 6

Factorization

(x² +9 ) to factorize the given problem

Solution:

The general form of the given equation is

(x² +a²) =(x+a) (x+a)

So. We factorize given problem

(x² +9 ) =(x+3) (x+3)

Greatest Common Divisor

Example:

12,4,36

We find the greatest common divisor of the given problems?

Solution:

We form the given problem

We divide by 4 all the numbers

12=2x2x 3

4=2x2 x 1

36=2x2x 9

We get the greatest common divisor of the given problems

Final answer is 2

Learning Perfect Number

Any number which is a positive integer  is called a perfect number if the sum of the factors(or divisors) of that number is equal to that number itself. Obviously, the factors excludes that number itself. In other words, we can say that a perfect number is an integer whose sum of factors(or divisors) is double the number.


Example of a perfect number

6

Factors of 6 are 1,2,3,6

Sum of the factors of 6(excluding 6) = 1+2+3 = 6 (the number itself)

Hence 6 is a perfect number.

Similarly, 28( factors of 28 excluding itself are 1,2,4,7 and 14, 1+2+4+7+14=28) is a perfect number.

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General form of a perfect number


The general form of a perfect number is given as 2p-1 (2p - 1), where p is a prime number. It is important to note that not all numbers of that form are perfect number but all perfect numbers will be of that form.

for p =2, 2p-1 (2p - 1) = 21 (22  - 1) =2*3 = 6, a perfect number.

Discrete Mathematics Relation

Introduction to discrete mathematics relation:

The discrete mathematics deals with functions and their properties, we noted the important property that all functions must have, namely that if a function does map a value from its domain to its co-domain, it must map this value to only one value in the co-domain.

Writing in set notation, if a is some fixed value:

` |{f(x)|x=a}|=1`

However, when we consider the relation, we relax this constriction, and so a relation may map one value to more than one other value. Having problem with Systems of Equations Solver keep reading my upcoming posts, i will try to help you.


Properties of discrete mathematics relation:-


In the following properties of discrete mathematics relation:-

Reflexive
Symmetric
Transitive
Antisymmetric
Trichotomy
Reflexive

Relation of the equality, = is reflexive. Examine that for all numbers a = a.So "=" is reflexive.

Symmetric

Relation is symmetric the values a and b:  a R b implies b R a.These type of relation is symmetric.

Transitive

Relation is transitive of all values of a, b, c: a R b and b R c implies a R c.These type of relation is transitive.

Antisymmetric

A relation is antisymmetric for all values a and b: a R b and b R a implies that a=b.These type of relation is antisymmetric

Trichotomy

A relation satisfy the all values a and b it holds true that: xRy or yRx.The  two relation numbers a and b, it is true that whether a ≥ b or b ≥ a (both if a = b).These type of relation is trichotomy. Please express your views of this topic How to Find Volume of a Cone by commenting on blog.


Example problems for discrete mathematics relation:-


Problem 1:-

`Let A = {1, 2, 3, 4, 5} and R : A harrA :-= {(a, b) : a |b}. What barR and R^-1?`

Solution:-

`{(1, 1) , (1, 2) , (1, 3) , (1, 4) , (1, 5) , (2, 2) , (2, 4) , (3, 3) , (4, 4) , (5, 5)}`

`barR = {(2, 1) , (2, 3) , (2, 5) , (3, 1) , (3, 2) , (3, 4) , (3, 5) , (4, 1) , (4, 2) , (4, 3) , (4, 5) , (5, 1) , (5, 2) , (5, 3) , (5, 4)}`

`R^-1 = {(1, 1) , (2, 1) , (3, 1) , (4, 1) , (5, 1) , (2, 2) , (4, 2) , (3, 3) , (4, 4) , (5, 5)}`

Problem 2:-

For each of the following relations of  pair which satisfies the relation ,and another pair which doesn't say whether each relation is reflexive relation,symmetric relation,transitive relation or anti-symmetric relation form discrete mathematics
(a) The relation on {1,2,3,4,5} defined by {(a,b)| a-b is even}
(b) The relation on {1,2,3,4,5} defined by {(a,b)| a+b is even}
(c) The relation on P, the set of all people,defined by {(a,b) | a and b have a common ancestor}

Solution:-

(a) The pair (4,2) satisfies the relation,(2,1) doesn't includes other relation.
(b) (3,3) satifies the relation,(3,2) doesn't includes other relation.
(c) (Bart,Lisa) satisfies the relation,(Homer,Marge) doesn't includes other relation.