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  • Match the following sets for all sets A, B and C ((A′ ∪ B′) – A)′  (a)    A – B [B′ ∪ (B′ – A)]′ (b)    A

Match the following sets for all sets A, B and C

  1. ((A' \cup B') - A)'          (a)    A - B
  2. [B'\cup (B'-A')]'          (b)    A
  3. (A_B) - (B-C)          (c)    B
  4. (A-B)\cap(C-B)   (d)    (A\times B)\cap(C\times B)
  5. A\times (B\cap C)                 (e)    (A\times B) \cup (A\times C)
  6. A\times (B\cup C)                 (f)    (A\cap C)-B

Answers (1)

(I) 

\begin{aligned} ((A' \cup B') - A)' &= [A'\cup B'\cap A]'& [\text{As }A-B = A\cap B'] \\ &= [A\cap B)'\cap A']' \\ &=[(A\cap B)']'\cup(A')' &[\text{As }(A')' =A] \\ &=(A\cup B)\cup A \\ &=A \end{aligned}

(II)

\begin{aligned} {[B'\cup (B'-A')]'} &=[B'\cup (B'\cap A')]' \\ &=(B')'\cap (B'\cap A')'\\ & = B\cap (B\cup A) \\ & = B \end{aligned} 

 

(iii)

 \begin{aligned} (A-B)-(B- C) &= (A\cap B')- (B\cap C') \\ & = (A\cap B') \cap (B\cap C')' \\ & = (A\cap B')\cap (B' \cap C) \\ & = [A\cap (B'\cap C)]\cap [B'\cap(B'\cap C)] \\ &=[A\cap (B'\cap C)]\cap B' \\ &= (A\cap B')\cap B' \\ &= (A\cap B')\\ &= A - B\end{aligned}

(iv)

 \begin{aligned} (A-B)\cap (C- B) &= (A\cap B')\cap(C\cap B') \\ & = (A\cap C)\cap B' \\ &= (A\cap C)- B\end{aligned}

(v) A\times (B\cap C) = (A\times B)\cap (A\times C)

(vi) A\times (B\cup C) = (A\times B)\cup (A\times C)

So (i) – b , (ii) – c, (iii) – a, (iv) – (f), (v) - (d), (vi) – (e) 

 

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