Could someone check this proof? If c\b subset c\a, then prove a subset b

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In summary, the conversation discusses the concept of subset in relation to a set of variables. It is mentioned that c\b is a subset of c\a, which means that if x is an element of c but not b, then it is also an element of c but not a. However, this is contradicted when x is an element of c and a, but not b. This leads to the conclusion that a must be a subset of b. The conversation also discusses negating the hypothesis and conclusion, with the conclusion being proven to be a contradiction. Ultimately, the participants reach an understanding and thank each other for their help.
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IntroAnalysis
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Assume c\b is a subset of c\a. This means if x Є c Λ (Not Є) b, then it is an Є c Λ (Not Є) a.

Assume x Є c Λ (Not Є) b, but is Not Є c Λ (Not Є) a. Then x Є c Λ a. But this contradicts,
c\b is a subset of c\a. Therefore, a must be subset of b.
 
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  • #2
It doesn't look right to me. Why would you negate the hypothesis ? This is not reductio ad absurdum.

Let's negate the conclusion: [itex] a\not\subset b [/itex], which means that [itex] \exists x\in a [/itex], so that [itex] x\not\in b [/itex]. But by hypothesis, [itex] \forall x\not\in b, x\not\in a [/itex]. Contradiction, right ?
 
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  • #3
You are correct. I see the difference. Thank you for the help.
 

1. What is the meaning of "subset" in this context?

In mathematics, the term "subset" refers to a set that contains all the elements of another set. In this proof, if c\b is a subset of c\a, it means that all the elements in c\b are also in c\a.

2. Can you provide an example to illustrate this proof?

Sure, let's say c = {1, 2, 3} and b = {1, 2}. Since b is a subset of c, we can rewrite it as b ⊆ c. Now, let's say a = {1, 3}. Since c\b = {3} and c\a = {2, 3}, we can see that c\b is a subset of c\a, as all the elements in c\b (which is only 3) are also in c\a (which is 2 and 3).

3. What is the significance of proving a subset relationship?

Proving a subset relationship between two sets is important in determining the relationship between their elements. In this proof, it shows that all the elements in c\b are also in c\a, which can help in simplifying further calculations or proofs.

4. Is this proof applicable in all cases?

Yes, this proof is applicable in all cases where the given sets meet the criteria of c\b being a subset of c\a. However, it is always important to double-check the proof for accuracy and to make sure that all the steps are logically sound.

5. Can you provide a step-by-step explanation of how to prove this statement?

Sure, here are the steps for proving a subset relationship like this:
1. Start by assuming that c\b is a subset of c\a (written as c\b ⊆ c\a).
2. Take an arbitrary element x from c\b and show that it is also in c\a.
3. Use the assumption c\b ⊆ c\a to show that x must be in c\a.
4. Repeat this process for all elements in c\b.
5. Conclude that since all elements in c\b are also in c\a, c\b is indeed a subset of c\a.
6. Therefore, the proof is complete and the statement is proven to be true.

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