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.

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.

 

[blue_raver22]

As a scientist, my response to this content would be to first clarify the purpose of providing counter examples in a proof. Counter examples are used to disprove a statement, which means that they show that the statement is not always true. In the case of set theory, counter examples are used to show that a particular equation or relationship between sets is not always valid.

In the example given, the statement being disproved is A - (B U C) = (A - B) U (A - C). This statement is claiming that the set A minus the union of sets B and C is equal to the union of A minus B and A minus C. To disprove this statement, we need to find a counter example, which means finding values for A, B, and C that make the equation false.

One way to do this is by using the method mentioned in the content, which is to choose specific values for A, B, and C and see if the equation holds true. However, this method may not always be efficient and may not always lead to a clear counter example.

Another method is to use Venn diagrams. Venn diagrams are useful tools for visualizing sets and their relationships. In the example given, we can draw a Venn diagram with three intersecting circles representing sets A, B, and C. We can then label the circles with the given values for A, B, and C (A = {a}, B = {a}, C = empty set).

From the Venn diagram, we can see that the left side of the equation (A - (B U C)) would be the shaded area outside of the union of B and C, which in this case is just the single element {a}. The right side of the equation ((A - B) U (A - C)) would be the union of the shaded areas outside of B and outside of C, which in this case would also be just the single element {a}. Therefore, the equation holds true in this case and we do not have a counter example.

However, as a scientist, it is important to also consider the possibility of other values for A, B, and C that could potentially lead to a counter example. In this case, we can see that if we change the value of C to {c} instead of an empty set, then the equation would not hold true. This is because the left side of the equation would still be just the single element