Solution to Formulate corresponding proof principles to prove the following properties about defined sets 1. A=B⇔A⊆B and … - Sikademy
Author Image

Archangel Macsika

Formulate corresponding proof principles to prove the following properties about defined sets 1. A=B⇔A⊆B and B ⊆ A 2. De Morgan’s Law by mathematical induction 3. Laws for three non-empty finite sets A, B, and C

The Answer to the Question
is below this banner.

Can't find a solution anywhere?

NEED A FAST ANSWER TO ANY QUESTION OR ASSIGNMENT?

Get the Answers Now!

You will get a detailed answer to your question or assignment in the shortest time possible.

Here's the Solution to this Question

1. Suppose we want to show A = B. If we show A ⊆ B, then every element of A is also in B, but there is still a possibility that B could have some elements that are not in A, so we can’t conclude A = B. But if in addition we also show B ⊆ A, then B can’t contain anything that is not in A, so A = B

2. Prove DeMorgan’s Law #1 Complement of the Union Equals the Intersection of the Complements

Let P = (A U B)' and Q = A' ∩ B' Let x be an arbitrary element of P then x ∈ P ⇒ x ∈ (A U B)' ⇒ x ∉ (A U B) ⇒ x ∉ A and x ∉ B ⇒ x ∈ A' and x ∈ B' ⇒ x ∈ A' ∩ B' ⇒ x ∈ Q

Prove DeMorgan’s Law #2 Complement of the Intersection Equals the Union of the Complements Let P = (A ∩ B)' and Q = A' U B' Let x be an arbitrary element of P then x ∈ P ⇒ x ∈ (A ∩ B)' ⇒ x ∉ (A ∩ B) ⇒ x ∉ A or x ∉ B ⇒ x ∈ A' or x ∈ B' ⇒ x ∈ A' U B' ⇒ x ∈ Q 

3. Using the Indirect Method

Let A , B , C be sets. If A ⊆ B and B ∩ C = ∅ , then A ∩ C = ∅

If we assume the conclusion is false and we obtain a contradiction --- then the theorem must be true.

Assume A ⊆ B and B ∩ C = ∅ , and A ∩ C ≠ ∅ . To prove that this cannot occur, let x∈A∩C.

x ∈ A ∩ C ⇒ x ∈ A and x ∈ C ⇒ x ∈ B and x ∈ C ⇒ x ∈ B ∩ C

But this contradicts the second premise. Hence, the theorem is proven.

Related Answers

Was this answer helpful?

Join our Community to stay in the know

Get updates for similar and other helpful Answers

Question ID: mtid-5-stid-8-sqid-3918-qpid-2617