Find the sets A and B if A – B = 1, 5, 7, 8, B

Find the sets A and B if A – B = {1, 5, 7, 8}, B – A = {2, 10}, and A ? B = {3, 6, 9}. Let A, B, and C be sets. Show that A) (A – B) – C ? A – C. Draw the Venn diagrams for each of these combinations of the sets A, B, and C. a) A ? (B ? C) b) A ? B ? C c) (A – B) ? (A – C) ? (B – C) Let Ai = {1, 2, 3, . . . , i} for i = 1, 2, 3, . . . Find: a) ? Ai (i=1 to n) b) ? Ai (i=1 to n)

Transcript

00:01 All right, so in this problem we've been given the difference and the intersection of two sets and been asked to find what those two sets are.

00:09 So let's first take a look.

00:10 We know that we're looking for set a and we know that we're looking for set b.

00:16 And we know that a intersect b is 3, 6, and 9.

00:21 So we know that a and b both contain 3, 6, and 9.

00:29 Then what we know here is that if we take a and remove b, this is what's going to be left.

00:38 So this is the result of a take away what they have in common.

00:43 So this is going to be what is in a.

00:47 And the same over here.

00:49 We're starting with b and we take away a.

00:51 So if we start with b and take away these things that they have in common, 2 and 10 we are getting left over.

01:00 So here are a and b.

01:03 And now we're asked to show that a minus b minus c is a subset of a c.

01:11 So let's start.

01:13 We know that a minus b minus c is going to be all of the x values where x is an element of a, but x is not an element of b.

01:28 And x can't be an element of c.

01:32 So that's how that translates.

01:34 But we also know that this is commutative.

01:37 We know that it doesn't matter what order we do these ands in.

01:42 So this says that x is an element of a and x is not an element of c.

01:48 It is both commutative and associative, and then x is not an element of b.

01:53 Well, what this says is that a minus c, minus b, and that would obviously be a subset of a minus c, because if we start with ac and we take some things out of there, that's still going to be a subset of b.

02:10 So what this is saying is if x is an element of a minus b, minus c, it is an element of a, and not an element of b of b.

02:37 Or c.

02:40 So we know then if x is an element of a minus b minus c then it's still going to be an element in a minus c and that's what we're trying to show.

03:11 And then we're asked to draw some venn diagrams.

03:15 And what i've done here is i've gone ahead and drawn some example venn diagrams of what these things look like...

Question

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