Question

Let A, B, and C be sets that are pairwise disjoint. (This means any pair of A, B, and C are disjoint. That is An B=), BAC =), and AnC = . Is it true that AUBUC| = A + B + C|? (Compute some examples before trying to prove this.)

Answer 1

Answer and Step-by-step explanation:

Given:

A, B and C are the pairwise disjoint sets

To prove:

|A + B+ C| = |A| +|B| +|C|

Proof:

Since, A, B and C are the pairwise disjoint sets, therefore all the possible pairs of A, B and C will also be disjoint and these are:

`A\cap B = \phi`;

`A\cap C = \phi`;

`B\cap C = \phi`;

`|A\cap B| = 0`;

`|A\cap C|= 0`;

`|B\cap C| = 0`;

Thus

`A\cap B\cap C = \phi`;

|A\cap B\cap C| = 0

Now, lets take an example:

Suppose, A = {b, c, d}

B = {e, f, g}

C = {h, i}

Now, we know that:

`n|A\cup B\cup C| = n|A| + n|B| + n|C| - n(A\cap B) - n(A\cap C) - n(B\cap C) - n|A\cap B\cap C|`

`|A\cup B\cup C| = {b, c, d, e, f, g, h, i}`

`n|A\cup B\cup C| = 8`

`n|A\cup B\cup C| = 3 + 3 + 2 = 8`

Thus

|A + B+ C| = |A| +|B| +|C|

Hence proved

Now, we know that:

`|A\cup B\cup C| = |A| + |B| + |C| - (A\cap B) - (A\cap C) - (B\cap C) - |A\cap B\cap C|`

from the above established data, we can write that:

`|A\cup B\cup C| = |A| +|B| +|C| - 0 - 0 - 0 - 0`

`|A\cup B\cup C| = |A| +|B| +|C|`

Hence, proved