Question

Prove that P∩Q=Q and P∩Q=P if P is a subset of Q.

Answer 1

1. Assume P is a subset of Q.

2. Let x be an arbitrary element in P∩Q, which means x belongs to both P and Q.

3. Since P is a subset of Q, every element in P is also an element of Q.

4. Therefore, x is an element of Q.

5. Since x was chosen arbitrarily, all elements in P∩Q hold.

6. Thus, P∩Q is a subset of Q.

Answer 2

To prove that P∩Q=Q, we need to show that every element in Q is also in P∩Q. Similarly, to prove that P∩Q=P, we need to show that every element in P is also in P∩Q.

Step-by-step explanation:

To prove that P∩Q=Q, we need to show that every element in Q is also in P∩Q. Since P is a subset of Q, every element in P is also in Q.

Therefore, every element in Q is in both P and Q, which means P∩Q=Q.

Similarly, to prove that P∩Q=P, we need to show that every element in P is also in P∩Q. Since P is a subset of Q, this means every element in P is in both P and Q. Therefore, P∩Q=P.