Question

Identify the correct steps involved in proving that if A, B, and C are sets, then |A ∪ B ∪ C| = |A| + |B| + |C| − |A ∩ B| − |A ∩ C| − |B ∩ C| + |A ∩ B ∩ C|.

Answer 1

Answer: The step-by-step proof is shown below.

Explanation:

To prove this, we need to first show that

|A U B| = |A| + |B| - |A n B|

Suppose A' and B' are the compliments of A and B respectively.

In a Venn diagram where set A and B intersects, we can see that

B = A' n B

So

|A U B| = |A U (A' n B)|

Where A and A' n B are mutually disjoint.

|A U B| = |A| U |A' n B|

= |A| + |A' U B|

= |A| + |B| - |A n B| (equation 1)

Now, to the main business.

Rewrite A U B U C to look like what we have proven above.

A U B U C = A U (B U C)

All we now have to do is prove for |A U (B U C)|

Easily from (equation 1)

|A U (B U C)| = |A| + |B U C| - |A n (B U C)|

= |A| + [|B| + |C| - |B n C|] - [(A n B) U (A n C)]

= |A| + [|B| + |C| - |B n C|] - [|A n B| + |A n C| - |(A n B) n (A n C)]|

= |A| + |B| + |C| - |B n C| - |A n B| - |A n C| + |A n B n C|

Because

(A n B) n (A n C) = A n B n C

Hence, the proof.

|A U (B U C)| = |A| + |B| + |C| - |B n C| - |A n B| - |A n C| + |A n B n C|

Answer 2

proved.

Explanation:

The detailed steps and appropriate values, elements of each sets were assumed to proved that the LHS is equal to the RHS as shown in the attached file.