# Proof: A Subset of B implies A/D Subset of B/D

• marinalee
In summary, to prove that if A is a subset of B then A/D is a subset of B/D, consider element x of A/D. This means x is an element of A and not of D. Since A is a subset of B, x is also an element of B. Therefore, combining these two facts, we can say that x is an element of B/D. This proves that A/D is a subset of B/D.
marinalee

## Homework Statement

Prove that if A is a subset of B then A/D is a subset of B/D.

## The Attempt at a Solution

Consider element x of A. Since A is a subset of B then for all x element of A, x is an element of B. Consider element x of A/D. If x is an element of D then x is not a member of A and thus it does not matter if x is an element of B. If x is not an element of D and is an element of A than x is also in B because x is an element of A. Thus, A/D is a subset of B.

Not even sure if this much is correct. How do I prove this basic "subtracting a set from both subsets" identity??

For one, you should not say "If x is an element of D then x is not a member of A ...", you should say "If x is an element of D then x is not a member of A/D", likewise for B/D.

But you take a lengthy confusing approach at this point. it would be more concise to say "Consider element x of A/D.." (here comes my part): x is an element of A/D implies x is in A and x is not in D...

Okay, I'll change that. But how do I finish the proof?

Pretty much like you did, I think, just with that shorter way of presenting it. Talk about why it implies x in is in B/D instead of considering B and D separately.

Oh! I think I get it. So the proof is: Consider x element of A/D. This means x is an element of A and not of D. This means x is a member of B, because all members of A are members of B, and since we already know that x is not an element of D we can combine these two facts and say x is an element of B/D.

It seems like you missed the point of considering x is in A/D instead of considering x is D and x is not in D.

You begin well here: "Consider element x of A. Since A is a subset of B then for all x element of A, x is an element of B. Consider element x of A/D." but then you begin taking cases of x in D or not in D. Don take cases. Just go straight into x is in A/D implies... whatever it implies. Leading into something about x is in B/D, right?

Um...I don't get it. :( What does A/D imply other than x is in A and not in D?

Oh, sorry, you're right. You're answer is good, I was thrown off by all the words. (we use all symbols in my class) :) Yay set theory *waves flag*

Perfect! Thanks! :D

## 1. What is set theory?

Set theory is a branch of mathematics that deals with the study of sets, which are collections of objects or elements. It provides a foundation for modern mathematics and is used to represent relationships between objects.

## 2. What is a proof in set theory?

A proof in set theory is a logical argument that uses axioms and rules of inference to demonstrate the truth of a statement about sets or their elements. It is used to show that a statement is true for all possible cases.

## 3. How do you prove equality of sets?

To prove equality of sets, you need to show that every element in one set is also in the other set, and vice versa. This can be done using the subset relation and the equality of cardinalities between the two sets.

## 4. What is the difference between a subset and a proper subset?

A subset is a set that contains all the elements of another set, while a proper subset is a subset that does not contain all the elements of the original set. In other words, a proper subset is a subset that is strictly smaller than the original set.

## 5. How do you prove a statement using mathematical induction?

To prove a statement using mathematical induction in set theory, you need to show that the statement is true for the smallest possible case (usually the empty set), and then show that if it is true for a particular set, it is also true for the next set. This process is repeated until the statement is shown to be true for all sets.

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