Set theory proof - counter examples

[amp92]

I'm having a problem with providing counter examples when disproving a statement. For example A - (B U C) = (A - B) U (A - C). The solution given was A = {a}, B = {a} and C = empty set.

My question is how can you work this out - i was told it's possible from the Venn diagrams but i'm not sure how this works. My method to find counter examples is usually to make A = {a}, B = {b} and C = {c} and then show the LHS doesn't equal the RIGHT. If it does i make changes to either A,B,C (i.e. use empty sets etc.). So for the example above can't you have A = {a}, B = {a} and C = {c}. How do you know C is an empty set?

Is it ok to stick with my method or can someone explain how to use the Venn diagrams for the LHS and RHS to find the counter examples.

Thank you :)

[I like Serena]

I'm having a problem with providing counter examples when disproving a statement. For example A - (B U C) = (A - B) U (A - C). The solution given was A = {a}, B = {a} and C = empty set.

My question is how can you work this out - i was told it's possible from the Venn diagrams but i'm not sure how this works. My method to find counter examples is usually to make A = {a}, B = {b} and C = {c} and then show the LHS doesn't equal the RIGHT. If it does i make changes to either A,B,C (i.e. use empty sets etc.). So for the example above can't you have A = {a}, B = {a} and C = {c}. How do you know C is an empty set?

Is it ok to stick with my method or can someone explain how to use the Venn diagrams for the LHS and RHS to find the counter examples.

Thank you :)

Here's a picture of the corresponding Venn diagrams.
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As you can see the diagrams are different in both cases.
In particular, if A ∩ B contains an element that is not part of C, we have a situation where the difference shows.
Let's say A ∩ B = {a} and {a} ⊄ C.
Then the simplest case of this would be if A=B={a} and C=Ø.

[amp92]

Ok i think i understand - you could have A=B={a} and C = {c} but it wouldn't be the simplest answer?

Also could you do exactly what you did for A ∩ B for A ∩ C instead as an alternative answer so the counter example would be A=C={a} and B = Ø?

Thank you so much for taking the time to explain this and for the diagrams :)

[I like Serena]

Ok i think i understand - you could have A=B={a} and C = {c} but it wouldn't be the simplest answer?

Yes, but your solution is fine too!
The counter example does not have to be the simplest possible, it just needs to do the job.
Of course, as a purist mathematician, I tend to search for the simplest most elegant solution.

Also could you do exactly what you did for A ∩ B for A ∩ C instead as an alternative answer so the counter example would be A=C={a} and B = Ø?

Thank you so much for taking the time to explain this and for the diagrams :)

Yes, that works just the same. :smile: