Prove set A is a subset of set B if and only if the intersection of A and B = A?

Question

If someone could explain this to me it would be of great help. How can I prove the following:

Set A is a subset of set B if and only if the intersection of A and B = A?

Answer 1

Since it is if and only if, it must be proven both ways.



First, need to prove A is a subset of B if the intersection of A and B = A. Assume that A is not a subset of B. Then, there is an element x in A that is not in B. Then, since x is not in B, the intersection of A and B will not include x. Thus, intersection of A and B ≠ A. But, we are given the intersection of A and B = A. Contradiction, so A is a subset of B.



Then, prove intersection of A and B = A if A is a subset of B. Assume the intersection of A and B ≠ A. Then, that means there exists a y in A such that y is not in B. But, this contradicts the given info that A is a subset of B. Contradiction, therefore the intersection A and B = A.



Since we've shown both directions, we are done!

Answer 2

The intersection of two sets contains only the elements that are common to both.



If x belongs to A and it belongs to B, then x is in the intersection.



If x belong to A but x does not belong to B, then x is NOT in the intersection.



"iff" is often used as a shortcut for "if and only if"

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To prove



A is a subset of B "iff" intersection of A and B = A



Method: for an iff, you must prove that it works both ways.



1) one way:

A is a subset of B therefore the intersection of A and B = A



A is a subset of B means that any element of A MUST also belong to B.

If x is in A, then it must also be in B.



The intersection of A and B will contain all the elements that are common to A and B.

Therefore, if x belongs to A and if A is a subset of B, then x must be common to both and it must belong to the intersection.



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2) the other way:

If the intersection of A and B = A, then A must be a subset of B.



Let x be an element of the intersection, and let the intersection be such that it is A. (meaning that there cannot be elements y such that they belong to A and are not in the intersection.



since x belongs to the intersection of A and B, then it must belong to both A and B (by definition of intersection.



Since every element of A must also belong to B (by saying that the intersection = A), then A must be a subset of B.



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Now, you must rewrite all this using the appropriate jargon and symbols.

Answer 3

If A is a subset of B, then x belongs to A ==> x belongs to B from the definition of a subset.

If every x is an element of A*B then every element x of A also belongs to B, since all points in any intersection of B with any other set also belongs to B. (Definition of intersection.)

Hence A is a subset pf B

Answer 4

Subset Proof

Answer 5

Suppose A∩B ≠ A. Then ∃ a ∈ A such that a ∉ B. ∴ A⊄B. Proof by contradiction.