EXAMPLE 1.1.3 SOLUTIONS

1.

This statement is true.

The expression on the left ("X complement") is the set of all elements that are in U but aren't in X.

On the other hand, W is the set of elements of U that are odd numbers. We see that this is exactly the same is X complement.

2.

This is true, because every element of Y is also an element of W.

3.

This is true, because every element of Y is also an element of W but Y is not equal to W.

4.

This is true, because every element of X is also an element of U. According to the definition of a Universal Set (U), this statement must be true for any set X.

5.

This is true. In order for this statement to be false, we would have to be able to find at least one entity that is an element of { } but isn't an element of E; we can't do so, since { } has no elements. We say that this statement is vacuously true.