Proving Set Equality: A-B, A∩B, B-A

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In summary, the identities involving complementation of sets are proven through the use of set algebra and logic. By applying de Morgan's laws and the distributive laws of union and intersection, the identities are simplified and shown to be equal to each other.
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simmonj7
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Homework Statement



Prove the following identities involving complementation of sets.
1. A ∩ [(A ∩ B)C] = A - B
2. A ∩ [(A ∩ BC)C] = A ∩ B
3. (A ∪ B) ∩ AC = B - A

Homework Equations


AC = A complement. If A is contained in some understood universal set U, then the complement of A is the set AC = {x ∈ U: x ∉ A}.
A - B = { x ∈ A and x ∉ B}
B - A = { x∈ B and x ∉ A}
A ∪ B = {x ∈ A or x ∈ B}
A ∩ B = { x ∈ A and x ∈ B}

The Attempt at a Solution


Since I am dealing with set equality and trying to show when one set equals another, I have been trying to show that x ∈ the first set if and only if x ∈ the second set in each of these cases.

I am not actually going to put this in legit proof form here but I will just summarize my steps for you and know that I plan on doing it better later once I figure it out entirely.

For part 1, I started off by choosing an arbitrary x in A ∩ [(A ∩ B)C]. Then, if x is in this set, x ∈ A and x ∉ A ∩ B. This means that x ∈ A and x ∉ A or x ∉ B. Well I am trying to show that this is equal to A - B, where x ∈ A and x ∉ B so I have the parts I need, however, I don't know what to do with the part where it says x ∉ A.

For part 2, I started off by choosing an arbitrary x in A ∩ [(A ∩ BC/SUP])C/SUP]]. Then, if x is in this set, x∈ A and x ∉ A ∩ BC/SUP]. This means that x ∈ A and x ∉ A or x ∉ B^C. Once again, this means that x∈A and x∉A or x∈ B. Well I am trying to show that this set is equal to A ∩ B, where x ∈ A and x ∈ B. But, once again, I have that extra part in the middle where x ∉ A that I don't know what to do about.

For part 3, I chose an arbitrary x in (A ∪ B) ∩ AC. Then, if x is in this set, x ∈ A or x ∈ B and x ∈ AC. Well, this means x ∈ A or x ∈ B and x ∉ A. This time, I have that other term in the front saying x ∈ A when I really need to get to B - A, where x∈ B and x ∉ A.

Am I completely missing something here?
Thanks.
 
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  • #2
That sort of proof gets tedious after while. Do you know deMorgan's laws applied to sets? http://planetmath.org/encyclopedia/DeMorgansLaws.html [Broken] and the distributive laws of union and intersection?
 
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  • #3
All we really know about de Morgan's laws is that, prior to this problem, we had to prove de Morgan's second law.
 
  • #4
simmonj7 said:
All we really know about de Morgan's laws is that, prior to this problem, we had to prove de Morgan's second law.

Well, ok then. For the first part if x is not in A, then x is not in A intersect anything, nor is it in A minus anything. So x is not in either side. So x is in one side iff x is in the other side. Because it's not in either. That's how the negative side of these things works. These things are a lot easier just using set algebra than juggling words. Hope you get to that phase soon.
 
  • #5
We are actually completely done with discussing sets so I don't know it we are going to get to there. :/
 
  • #6
Too bad, maybe next course? Here's the first one. An(AnB)^C=An(A^C U B^C)=(An(A^C))U(An(B^C)=An(B^C), since An(A^C) is empty. And An(B^C)=A-B. Done.
 
  • #7
That actually makes some sense to me but I wonder if I will even be away to get away with set algebra since we haven't covered it.
But thank you so much for that.
 
  • #8
simmonj7 said:
That actually makes some sense to me but I wonder if I will even be away to get away with set algebra since we haven't covered it.
But thank you so much for that.

You can do it just by logic too. Take the first one, you got up to "This means that x ∈ A and x ∉ A or x ∉ B." If x is in A, then x can't NOT be in A too. So the second part of the 'or' must be true. So you get x ∈ A and x ∉ B. Which is A-B.
 
  • #9
Thank you.
It's all solved now.
:)
 

1. What is the definition of set equality?

Set equality is a mathematical concept that refers to the relationship between two sets, where both sets have exactly the same elements. In other words, if all the elements in set A are also present in set B, and vice versa, then set A and set B are considered equal.

2. What is the difference between set difference and set intersection?

Set difference, denoted by A-B, refers to the elements that are present in set A but not in set B. On the other hand, set intersection, denoted by A∩B, refers to the elements that are common to both sets A and B.

3. How do you prove set equality?

To prove that two sets are equal, you must show that they have the same elements. This can be done by showing that set A is a subset of set B and set B is a subset of set A. Alternatively, you can show that A-B and B-A are both empty sets, which indicates that the two sets have the same elements.

4. Can two sets be equal if they have different cardinalities?

No, two sets cannot be equal if they have different cardinalities. Cardinality refers to the number of elements in a set, and if two sets have different cardinalities, it means they have different numbers of elements and therefore cannot be equal.

5. Is set equality the same as set equivalence?

No, set equality and set equivalence are not the same. Set equality, as mentioned earlier, refers to the relationship between two sets that have exactly the same elements. Set equivalence, on the other hand, refers to the relationship between two sets that have the same elements and the same cardinality.

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