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(A ∩ BLet A, B, and C be three sets. Prove that A-(BUC) = (A-B) ∩ (A-C)

Solution)

L.H.S = A - (B U C)

A ∩ (B U C)^{c}

A ∩ (B^{c}∩ C^{c})

(A ∩ B^{c}) ∩ (A∩ C^{c})

(AUB) ∩ (AUC)

R.H.S = (A-B) ∩ (A-C)

(A∩B^{c}) ∩ (A∩C^{c})

(AUB) ∩ (AUC)

L.H.S = R.H.S

Is this correct?

A ∩ B

In fact

(A ∩ B

- Apr 13, 2013

- 3,718

We should work on

Let [tex]A, B, C[/tex] be three sets.

Prove that:.[tex]A - (B \cup C) \:=\: (A-B) \cap (A-C)[/tex]

[tex]\begin{array}{cccccc}

1. & A -(B \cap C) && 1. &\text{Given} \\

2. & A \cap(B\cup C)^c && 2. &\text{def. Subtr'n} \\

3. & A \cap B^c \cap C^c && 3. & \text{DeMorgan} \\

4. & A \cap A \cap B^c \cap C^c && 4. & \text{Duplication} \\

5. & A\cap B^c \cap A \cap C^c && 5. & \text{Commutative} \\

6. & (A \cap B^c) \cap (A \cap C^c) && 6. & \text{Associative} \\

7. & (A-B) \cap (A-C) && 7. & \text{def. Subtr'n}\end{array}[/tex]