# Prove (A∩B)C=AC∩BC is FALSE: Counterargument Needed

• Blue_Wind
In summary, the conversation discussed the task of proving the statement (A∩B)C = AC∩BC is false. The homework equations suggested trying to prove the statement to find a problem and come up with a counterargument. The attempt at a solution involved proving one part of the statement to be true, but the second part was proving to be difficult. The professor advised to simply come up with a counterexample instead of trying to prove the statement false. The suggestion to use a Venn diagram to construct a counterexample was given.
Blue_Wind

## Homework Statement

A,B and C are sets.
Prove (A∩B)C = AC∩BC is FALSE
That is, I have to give a counterargument for this statement.

## Homework Equations

I can't find a counterargument directly. My professor suggest trying to prove the statement to find a problem and come up with the counterargument.
To prove this is false, first must prove that
AC∩BC$$\subseteq$$(A∩B)C is false, OR
(A∩B)C$$\subseteq$$AC∩BC is false.

## The Attempt at a Solution

I have proven (A∩B)C$$\subseteq$$AC∩BC is true by:
• w is a string
• Let w$$\in$$(A∩B)C then $$\exists$$u$$\in$$(A∩B)and $$\exists$$v$$\in$$C where w=uv
• If $$\exists$$u$$\in$$A then w=uv$$\in$$AC and $$\exists$$u$$\in$$B and w=uv$$\in$$BC
• Hence (A∩B)C$$\subseteq$$AC∩BC

However, I wasn't able to prove AC∩BC$$\subseteq$$(A∩B)C is false.

• w is a string
• Let w$$\in$$AC and w$$\in$$BC
• Then $$\exists$$u$$\in$$A and $$\exists$$v$$\in$$C where w=uv
• Also $$\exists$$u$$\in$$B and $$\exists$$v$$\in$$C where w=uv
• Then $$\exists$$u$$\in$$A∩B and $$\exists$$v$$\in$$C
• Hence AC∩BC$$\subseteq$$(A∩B)C ?

My professor said that the second part is wrong, but I have already tried over an hour but still can not make the second part false nor just come up with a counterargument.

I'm really not good with logic, can anyone help me?
I still have a lot of programming assignment waiting for me to do.

you don't really need to prove anything..
you just have to come up with a counterexample

If you draw a Venn diagram of the 3 sets you can construct your counter example

Good luck!

Counterargument:

The statement (A∩B)C=AC∩BC is not always false. In fact, it can be true in certain cases. For example, let A={1,2,3}, B={2,3,4}, and C={3,4,5}. Then (A∩B)C={(1,2,3), (2,3,4), (3,4,5)} and AC∩BC={(1,2,3), (2,3,4), (3,4,5)}. In this case, the statement is true. Therefore, the statement cannot be proven to be always false and a counterargument cannot be found.

## 1. What is the definition of (A∩B)C and AC∩BC?

The notation (A∩B)C represents the set of elements that are in both A and B, and then in turn, also in C. On the other hand, AC∩BC represents the elements that are in A and C, and also in B and C.

## 2. Why is the statement (A∩B)C=AC∩BC false?

This statement is false because the order of operations matters when dealing with set operations. In this case, (A∩B)C means that we first find the intersection of A and B, and then find the intersection of that result with C. On the other hand, AC∩BC means that we first find the intersection of A and C, and then find the intersection of that result with B and C. These two operations will not always give the same result, thus making the statement false.

## 3. Can you provide an example to illustrate the falsity of (A∩B)C=AC∩BC?

Yes, let's consider the following sets: A = {1, 2, 3}, B = {2, 3, 4}, and C = {3, 4, 5}. Using the definition of (A∩B)C, we get (A∩B)C = (2, 3)C = ∅, since there are no elements that are in both (2, 3) and C. However, using the definition of AC∩BC, we get AC∩BC = (1, 3)∩(3, 4) = {3}, since 3 is the only element that is in both (1, 3) and (3, 4).

## 4. Is it possible to prove that (A∩B)C=AC∩BC using a counterargument?

No, it is not possible to prove the statement using a counterargument. A counterargument is used to disprove a statement by providing a specific example or reasoning that contradicts the statement. In this case, we can use a counterexample to show that the statement is false, but we cannot prove it to be true using a counterargument.

## 5. How can we correct the statement to make it true?

In order to make the statement true, we need to change the order of operations. Instead of (A∩B)C=AC∩BC, we can write (A∩C)B=AB∩CB. This statement is true because we are now finding the intersection of A and C first, and then finding the intersection of that result with B. This is the correct order of operations for the given sets.

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