math 189 in class work

1. In Class Notes: Wednesay, 26 Feb 2014
Tyler Murphy
February 26, 2014
Let’s take a look at the conjecture made in class today about problem 2.2.19a.
We are given that A ∩ B ⊆ C and Ac ∩ B ⊆ C.
We want to show that this means that B ⊆ C.
The question that arose was about the assumption. Can we claim that:
(A ∩ B) ∩ (Ac ∩ B) ⊆ C?
More importantly, does this prove that B ⊆ C?
Let me just say: Yes. To both. Now let’s prove it.
First, we can show that
(A ∩ B) ∩ (Ac ∩ B) ⊆ C.
Then we will show that
(A ∩ B) ∩ (Ac ∩ B) = B.
Ask yourself why these two things will give us that B ⊆ C..
Let’s look at the first claim.
We are given that A ∩ B ⊆ C and Ac ∩ B ⊆ C. WTS that (A ∩ B) ∩ (Ac ∩ B) ⊆ C.
Proof. First, suppose that we can combine A ∩ B and Ac ∩ B into (A ∩ B) ∩ (Ac ∩ B) since
”and” is similiar to ∩.
Now we want to show that (A ∩ B) ∩ (Ac ∩ B) ⊆ C.
Since the intersection of two sets is a subset of each of those sets, we have:
(A ∩ B) ∩ (Ac ∩ B) ⊆ (A ∩ B) ⊆ C.
and
(A ∩ B) ∩ (Ac ∩ B) ⊆ (Ac ∩ B) ⊆ C.
So (A ∩ B) ∩ (Ac ∩ B) ⊆ C.
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2. Now let’s look at the other claim. That (A ∩ B) ∩ (Ac ∩ B) = B.
Proof.
(A ∩ B) ∩ (Ac ∩ B).
Since intersection is associative, we can drop the parenthesis.
A ∩ B ∩ Ac ∩ B.
Further, because X ∩ Y = Y ∩ X (commutativity), we can write:
A ∩ Ac ∩ B ∩ B.
So,
(A ∩ Ac ) ∩ (B ∩ B).
Since A ∩ Ac = ∅ and B ∩ B = B (by previous proofs), we have:
(∅) ∩ (B).
Which is:
B.
So,
(A ∩ B) ∩ (Ac ∩ B) = B.
So, through ALL of this mess, we have proven that if A ∩ B ⊆ C and Ac ∩ B ⊆ C, then
B ⊆ C.
Try to understand what we did here. This occurred in two parts. First, we had
to show that our conjecture was true. Namely, that A ∩ B ⊆ C and Ac ∩ B ⊆ C ⇒
(A ∩ B) ∩ (Ac ∩ B) ⊆ C.
Then we had to link our conjecture to the original problem. We needed to show that
(A ∩ B) ∩ (Ac ∩ B) = B.
Since we were able to do that, we have completed the overall proof.
Note: All of the proofs that I design for these notes are done in a lot of detail. Sometimes,
you won’t need that much to complete the proof. Also note that these proofs can be done
LOTS of ways. If you prove something in a different manner than I have (like contradiction,
contrapositive, or a different manner directly), feel free to email it to me and I’ll write it
up and post it for everyone. The more ways we can see something done, the better we can
learn it.
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