Perfect Difference Set

Introduction to perfect difference set:

Collections of an element are called as set. Elements of a set are also called as objects of a set or numbers of a set. Set subtraction is one of the operations of a set. Set subtraction is also called as difference set. If the result of difference set has only positive integers then they are called as perfect difference set.  Let us see perfect difference set in this article.

Perfect Difference Set:

Difference set:

Set subtraction is one of the other names of difference set. Let us consider ‘A’ and ‘B’ are given set. The set of all elements of A which is not present in set B is called as A – B. The elements of difference set are both negative integer and non - negative integer.

Example:

A = {-2, -1, 1, 2, 3, 4, 5}

B = {1, 2, 3}

A – B = {-2, -1, 4, 5}

Perfect difference set:

If the difference set have non-negative integers only – called as perfect difference set.

Example:

A = {1, 2, 3, 4, 5}

B = {1, 2, 3}

A – B = {4, 5}

The result of A – B is not containing the negative integers so the above difference set is a perfect difference set.

Problems for Perfect Difference Set:

Problem 1:

A = {1, 3, 4, 8, 12, 15, 16, 20, 24, 32, 44} B = {1, 4, 8, 15, 20, 22, 30, 41, 44}

Find A – B and B – A. Also state it is a proper difference set or not.

Solution:

Given

A = {1, 3, 4, 8, 12, 15, 16, 20, 24, 32, 44}

B = {1, 4, 8, 15, 20, 22, 30, 41, 44}

A – B = {3, 12, 16, 24, 32}

B – A = {22, 30, 41}

A – B and B – A have only non-negative integers so A – B and B – A are perfect difference set.

Problem 2:

A = {-2, -6, 16, 23, 25, 28, 29, 30} B = {-1, -3, -6, 16, 25, 28, 30}

Find A – B and B – A

Solution:

Given

A = {-2, -6, 16, 23, 25, 28, 29, 30}

B = {-1, -3, -6, 16, 25, 28, 30}

A – B = {-2, 23, 29}

B – A = {-1, -3}

A – B and B – A have negative integers also therefore A – B and B – A are not a perfect difference set.