Let X, Y and Z be sets. For each of the following statements, prove it or give a counterexample.(a) If X is not a subset of Y , then Y is a subset of X.(b) (X − Y ) − X = ∅.(c) X ∪ (Y − Z) = (X ∪ Y ) − (X ∪ Z).

The statement which is true is (c) X ∪ (Y − Z) = (X ∪ Y ) − (X ∪ Z) and the

statements which are false are a) If X is not a subset of Y , then Y is a subset of X and (b) (X − Y ) − X = ∅.

The statement "If X is not a subset of Y, then Y is a subset of X" is false. A counterexample is sufficient to disprove this statement.

Let's consider X = {1, 2} and Y = {2, 3}. X is not a subset of Y because it contains the element 1 which is not in Y.

However, Y is not a subset of X either because it contains the element 3 which is not in X. Therefore, the statement is false.

The statement "(X - Y) - X = ∅" is false. To prove this, we need to find a counterexample.

Let's consider X = {1, 2, 3} and Y = {2, 3}. The set (X - Y) - X can be computed as ({1} - {2, 3}) - {1}, which simplifies to the empty set ∅. However, the statement claims that (X - Y) - X should be equal to ∅, which is false in this case. Therefore, the statement is false.

The statement "X ∪ (Y - Z) = (X ∪ Y) - (X ∪ Z)" is true. To prove this, we need to show that the sets on both sides of the equation contain the same elements.

Let's consider an arbitrary element x.

If x is in X ∪ (Y - Z), it means x is either in X or in (Y - Z). If x is in X, then it is also in X ∪ Y and X ∪ Z, so it will be in (X ∪ Y) - (X ∪ Z). If x is in (Y - Z), it is not in Z, so it will be in X ∪ Z. Therefore, x is in (X ∪ Y) - (X ∪ Z).

Conversely, if x is in (X ∪ Y) - (X ∪ Z), it means x is in X ∪ Y but not in X ∪ Z. This implies that x is either in X or in Y but not in Z. Therefore, x will be in X ∪ (Y - Z).

Since we have shown that an arbitrary element x is in both X ∪ (Y - Z) and (X ∪ Y) - (X ∪ Z), we can conclude that the two sets are equal. Hence, the statement is true.

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