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Hello everyone. In this video, we're 
going to do another if and only if proof.

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This time we're going to use sets.

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Prove that if…

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or prove that A is a subset 
of B, subset or equal to B,

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if and only if A intersect B is equal to A.

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So again with an if and only 
if proof, we're going to prove 

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that the hypothesis implies the conclusion, 
and that the conclusion implies

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the hypothesis. So the forward
direction and the reverse direction.

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For the forward direction,

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assume

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that A is a subset or equal to B.

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Now we're trying to prove a set equality

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and remember to prove a set equality, 
you need to show that every element

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in the left-hand set is in the right-hand set, 
and every element in the right-hand set

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is indeed a member of the left-hand set.

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So to show…

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what am I doing? So to show

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A intersect B

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is a subset of A, we note

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that A intersect B is the set

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of all elements

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contained

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in both

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A and B,

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and so

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all elements

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are contained

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in A.

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That's strictly by the definition. 
If you're in both A and B,

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then you're most certainly inside A. So that's true,

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A intersect B is definitely 
a subset or equal to A.

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To show

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A is a subset of A intersect B,

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we're going to do this by showing 
an arbitrary element of A 

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must be inside A intersect B.

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So I'm going to say let x be an element of A.

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You could have done this for the other part as
well, but the other part really does just follow

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by writing out the sentence of what the set is.

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So I'm going to take this set - I'm 
going to take an arbitrary element in A,

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and I'm going to show that 
it lives in this right-hand set. 

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So how do I do that? I can't do that on 
its own because on its own that's not true,

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but with the hypothesis I can get there.

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So this statement was true no
matter what the hypothesis was,

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but this statement we need to use hypothesis.

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Then since

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A is contained inside B, we have

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that

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x is an element of B.

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As x is inside A 

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and x is inside B, we have

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that x is inside A intersect B, by definition. 

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Remember this is the set of all
elements that are in A and B.

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Well x is in A and B, so we're done.

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So since

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A intersect B is a subset or equal to A

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and A is a subset of A [intersect] B, 

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we have that

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A is equal to A intersect B, alright.

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I guess I wrote it in the flipped 
direction, but that's fine. Symmetry

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of equality, that's still okay.

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Alright, so the forward direction
we're okay. For the reverse direction,

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we're going to assume

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that A intersect B is equal to A.

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Alright, and so what are we trying to show? 
We're trying to show therefore that

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A is a subset or equal to B.

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Let's look at this set right here A intersect B.

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Notice that before we 
showed that A intersect B

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is a subset or equal to A, right, 

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and we use nothing about 
the hypothesis. So in fact,

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this same argument is going to hold to show
that A intersect B is a subset or equal to B.

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So remember as mathematicians, 
we try to be as lazy as possible.

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Sort of a fun little joke,

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but we've already really done 
the same amount of work here.

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So really all I have to do is just copy the
work except I change this A to a B.

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So I'm going to say as before,

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we have that

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A intersect B is a subset or equal to B

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since...

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since A intersect B is all elements 

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contained

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in both A and B, alright. 

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Great, so that's true.

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As A is equal to A intersect B,

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for any x inside A,

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x is inside A intersect B

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and since…

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oops, and since A intersect B is
contained inside B, we have that

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x is an element of B.

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So we've shown that any
arbitrary element of A

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must live inside B,

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that's precisely the - that's precisely the 
statement that A is contained inside B.

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So therefore,

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always start sentences with words don't start
sentences with symbols, it's bad form.

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There we go. A is inside B.

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That's it, that's a full if and 
only if proof using set notation.

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So hopefully this helps and all the best.