Solving Set Equality Proof Homework

In summary, the conversation discusses the problem of showing that the sets (A-B)∪C and (A∪C)-(B∪C) are equal. The speaker begins by explaining the general method of showing two sets to be equal, by proving that each is a proper subset of the other. They then analyze the left hand side of the equation and point out a potential flaw in the argument. Another speaker suggests drawing a Venn diagram and the first speaker realizes their mistake. The conversation concludes with the decision to email the professor about the problem.
  • #1
Haydo
20
0

Homework Statement



Let ##A, B, C## be sets with ##A \subseteq B##. Show ##(A-B)\cup C=(A\cup C)-(B\cup C)##

Homework Equations



None.

The Attempt at a Solution



So, generally, one shows two sets to be equal by showing that each is a proper subset of the other. I started with the LHS. Thus, if x is in (A-B)UC, x is in (A-B) or x is in C. But if x is in C, then x is not in RHS. So it seems that the expression does not hold. Am I thinking of this wrong, or did the assignment writer make an error?
 
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  • #2
If x is in C, then x can still be in the right hand side. It might help to draw a Venn diagram.
 
  • #3
I should say, consider the case where x is in C and x is not in (A-B). Then x cannot be in (AUC)-(BUC), right?
 
  • #4
You know what, I think you're right! If x is in C, then for x to be in the RHS it could not be in ## B \cup C ##. But then it cannot be in ## C ##. Also, my first Venn diagram had a mistake in it.
 
  • #5
Hm. That's what I thought. I'll email the professor then. Maybe he was just trying to see if we were paying attention, but the problem definitely said to prove the equality, not to prove it or show it is false.
 

1. How do I know if two sets are equal?

In order for two sets to be equal, they must have the same elements. This means that every element in one set must also be present in the other set, and vice versa. If this condition is not met, the sets are not equal.

2. What is the process for solving set equality proofs?

To solve set equality proofs, you must first determine the elements in each set and then use logical reasoning to show that they are equal. This often involves using set operations, such as union, intersection, and complement, to manipulate the sets and demonstrate their equality.

3. Can I use diagrams or visual aids to solve set equality proofs?

Yes, diagrams and visual aids can be helpful in understanding and solving set equality proofs. They can provide a visual representation of the sets and their elements, making it easier to see how they are related and how they can be manipulated to show equality.

4. What are some common mistakes to avoid when solving set equality proofs?

One common mistake is assuming that two sets are equal without fully understanding the elements in each set. It's important to carefully examine the elements and use logical reasoning to prove equality. Another mistake is using incorrect set operations or not fully understanding how to use them.

5. Are there any tips for solving set equality proofs efficiently?

One tip is to start by writing out the elements in each set and then using set operations to manipulate them. It can also be helpful to work backwards and start with the desired outcome (the two sets being equal) and then figure out the steps needed to get there. Additionally, practicing and becoming familiar with set operations can make solving set equality proofs more efficient.

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