Need help with this math problem?

Question

a) Prove that if E,F are sets then E∩(E∪F)=E and E∪(E∩F)=E.

b) Suppose that A,B,C are sets satisfying A∩B=A∩C and A∪B=A∪C.
Prove that B=C.

c) Show that only assuming one of the conditions in the previous part is not sufficent to prove that B=C. In other words, give examples (one for each condition) where one condition holds and yet B≠C.

? · Accepted Answer

(a)
i) Prove E∩(EUF)=E

Let EUF = A

  E∩(EUF)
    = E∩A

since E is a subset of A

  E∩A = E

Hence

  E∩(EUF) = E


ii) Prove EU(E∩F)=E

Let E∩F = B

Since B is a subset of E

  EUB = E

Hence

  EU(E∩F)=E


(b) A∩B=A∩C and AUB=AUC. Prove that B=C.

Let A∩¬B = E

  A∩¬C = E

For AUB=AUC

  ¬E∩AUB=¬E∩AUC

  B = C


(c)
Let A = {1,2,3}, B={2,4}, C={2,5}

Then

  A∩B = {2}
  A∩C = {2}

Thus A∩B=A∩C holds but B ≠ C

Let A = {1,2,3}, B={2,4}, C={3,4}

Then

  AUB = {1,2,3,4}
  AUC = {1,2,3,4}

Thus AUB=AUC holds but B ≠ C

Trending News

Need help with this math problem?

1 Answer