Answer to Question #86929 in Algebra for sonali mansingh

Question

Let A = { x ∈Z x is a multiple of 5} and B = { x∈Z x is a divisor of 20}.
Represent B,A and A &cap B c, where &cap means the intersection of sets and B c means the complement of the set B,
by the listing method and in a Venn diagram.

Answer 1

The set A can be rewritten using the definition of multiple as follows: 

$$A = \left\{ 5x \mid x\in\mathbb{Z}\right\}.$$

Set B consists of 6 elements because 20 has 6 divisors and $20=2^2*5$. Because of that:

$$B=\left\{1,2,4,5,10,20\right\}.$$

Since the universe in our case is integer numbers, then $B^{c}=\mathbb{Z}\setminus B =\mathbb{Z}\setminus \left\{1,2,4,5,10,20\right\}$, it means that $B^c$ contains all integer numbers beside 1, 2, 4, 5, 10, 20. Then the intersection of these sets will be the following set: 

$$A\cap B^c=\left\{ 5x \mid x\in \mathbb{Z}, x = \not 1,2,4\right\}.$$

The second condition occurs from the fact that 5=5*1, 10=5*2, 20=5*4 are not included in the set $B^c$ , thus cannot be in the intersection.

Venn diagram: