Final answer:
To prove that a×(b∩c) = (a×b)∩(a×c), we can show that each element in a×(b∩c) is also in (a×b)∩(a×c), and vice versa.
Step-by-step explanation:
To prove that a×(b∩c) = (a×b)∩(a×c), we need to show that each element in a×(b∩c) is also in (a×b)∩(a×c), and vice versa.Let (x, y) be an element in a×(b∩c). This means that x is in set a, and y is in the intersection of sets b and c. Therefore, (x, y) is also in (a×b) and (a×c). Hence, a×(b∩c) is a subset of (a×b)∩(a×c).To show the reverse inclusion, let (x, y) be an element in (a×b)∩(a×c). This means that (x, y) is in both (a×b) and (a×c).
From this, we can conclude that x is in set a, and y is in both set b and set c. Therefore, (x, y) is also in a×(b∩c). Hence, (a×b)∩(a×c) is a subset of a×(b∩c).Since both sets are subsets of each other, we can conclude that a×(b∩c) = (a×b)∩(a×c).