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

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).