Question

If A={factors of 6}, B={factors of 8}, and C={factors of 9}, prove that: (A∩B)∩C = A∩(B∩C)

Answer 1

To prove (A∩B)∩C = A∩(B∩C), we need to show that both sides of the equation represent the same set. First, we find (A∩B)∩C by taking the common factors of the sets. Then, we find A∩(B∩C) using the same approach. By comparing the two sets, we can see that they are both equal to {1}, thus proving the equality.

Step-by-step explanation:

To prove (A∩B)∩C = A∩(B∩C), we need to show that both sides of the equation represent the same set.

Let's start by finding (A∩B)∩C:

1. Find A∩B by taking the common factors of 6 and 8. The factors of 6 are 1, 2, 3, and 6, while the factors of 8 are 1, 2, 4, and 8. The common factors of 6 and 8 are 1 and 2, so A∩B = {1, 2}.
2. Now, find (A∩B)∩C by taking the common factors of A∩B and 9. The factors of 9 are 1, 3, and 9. The common factors of {1, 2} and 9 are 1, so (A∩B)∩C = {1}.

Next, let's find A∩(B∩C):

1. Find B∩C by taking the common factors of 8 and 9. The factors of 8 are 1, 2, 4, and 8, while the factors of 9 are 1, 3, and 9. The common factors of 8 and 9 are 1, so B∩C = {1}.
2. Now, find A∩(B∩C) by taking the common factors of A and B∩C. The factors of 6 are 1, 2, 3, and 6, and the common factor with {1} is 1, so A∩(B∩C) = {1}.

Since (A∩B)∩C = A∩(B∩C) = {1}, we have proved that (A∩B)∩C = A∩(B∩C).