Question

Suppose n(U) = w, n(A) = x, n(B) = y, and n(A ∪ B) = z. (a) Why must x be less than or equal to z? A ⊆ (A ∪ B) (A ∪ B) ⊆ A     (A ∪ B) ⊆ B A ⊆ (A ∩ B) (A ∩ B) ⊆ A

Answer 1

First, note some definitions:

1. A set is a collection of distinct objects, considered as an object in its own right.

2. If A is a set, then n(A) is a number of elements in A.

3. Set X is a subset of a set Y if and only if every object of X is also an object of Y. Here use notation X⊆Y.

4. A set X∪Y is called union of sets X and Y. This is the set of all distinct elements that are in X or Y.

Now remember that

• A⊆A∪B (set A∪B consists of all elements of set A and of all elements of set B);
• B⊆A∪B (set A∪B consists of all elements of set A and of all elements of set B);
• the number of elements in subset is always less than or equal to the number of elements in set.

Then n(A)≤n(A∪B) (or x≤z), because A ⊆ (A ∪ B).