Answer to Question #282380 in Discrete Mathematics for islam

Question #282380

4. Let A and B be sets. Prove the commutative laws from Table 1 by showing that

a) A ∪ B = B ∪ A.

b) A ∩ B = B ∩ A.


1
Expert's answer
2021-12-24T14:23:27-0500

a) Let xAB.x∈A∪B. Then xAx∈A or xB.x∈B. Which implies  xBx∈B or xA.x∈A.

Hence xBA.x∈B∪A.

Thus ABBA.A∪B⊆B∪A.

Similarly, we can show that BAAB.B∪A⊆A∪B.

Therefore, AB=BA.A∪B=B∪A.


b) Let xAB.x∈A∩B. Then xAx∈A and xB.x∈B. Which implies xBx∈B and xA.x∈A. Hence xBA.x∈B∩A.

Thus ABBA.A∩B⊆B∩A.

Similarly, we can show that BAAB.B∩A⊆A∩B.

Therefore, AB=BA.A∩B=B∩A.



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