Sets

Let A and B be sets. Prove each of the following statements. Remember, using a Venn diagram might help to visualize the problem, but it's not a proof!

a. (A ∩ B) ⊆ A
b. A ⊆ A ∪ B
c. A ∩ (B - A) = 0

a) (A ∩ B) ⊆ A

Let x be in (A ∩ B), then x is in A and x is in B. Hence x is in A. Thus (A ∩ B) ⊆ A

b) A ⊆ A ∪ B

This one seems so easy to prove, I'm not sure what else to say about it:

Let x be in A. If x is in A then x is in A ∪ B. Thus A ⊆ A ∪ B

c) A ∩ (B - A) = 0

(B - A) = {x | x is in B and x is not in A}. So intuitively this equality should make sense. You take everything that is in A and intersect with everything that is in B takeaway A, so A and (B-A) should be mutually disjoint sets).

For some reason I can't think of how to prove this one, sorry. Someone else will have to tackle it.

Quote:

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Originally Posted by captainjapan

Let A and B be sets. Prove each of the following statements. Remember, using a Venn diagram might help to visualize the problem, but it's not a proof!

a. (A ∩ B) ⊆ A
b. A ⊆ A ∪ B
c. A ∩ (B - A) = 0

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a) xε( Α B) <====> (xεA & xεB) ====> xεA.


b) xεA =====> xεA v xεB <=====> xε( AUB)


c) xε[A (B-A)] <====> xεA & (xεB & ~xεA) <=====>


(xεA & ~xεΑ) & xεB <====> xε ( Α A') & xεB <=====> xε( Φ B) <====> xεΦ.


Because A A' = Φ and , Φ B =Φ

Φ being the empty set