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’.
I have recently faced lot of problem while learning Area of an Equilateral Triangle, But thank to online resources of math which helped me to learn myself easily on net.
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 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.
I like to share this Sum of Harmonic Series with you all through my article.
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’.
I have recently faced lot of problem while learning Area of an Equilateral Triangle, But thank to online resources of math which helped me to learn myself easily on net.
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.
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