# Thread: Question on set operations

1. ## Question on set operations

is this statement true?

(A - B) - C = (A - C) - (B - C)

2. Hello, brudman!

Is this statement true?

. . $\displaystyle (A - B) - C \:=\: (A - C) - (B - C)$ . . . . . yes
Definition: .$\displaystyle P - Q \:=\:P \cap Q'$

The left side is:

. . $\displaystyle \begin{array}{cc} (A - B) - C & \text{Given} \\ (A \cap B') - C & \text{d{e}f.Subtr'n}\\ (A \cap B') \cap C' & \text{d{e}f.Subtr'n} \\ A \cap B' \cap C' & \text{Associative} \end{array}$

The right side is:

. . $\displaystyle \begin{array}{cc} (A-C) - (B-C) & \text{Given} \\ (A \cap C') - (B \cap C') & \text{d{e}f.Subtr'n} \\ (A \cap C') \cap (B \cap C')' & \text{d{e}f.Subtr'n} \\ (A \cap C') \cap (B' \cup C) & \text{DeMorgan} \\ A \cap C' \cap (B' \cup C) & \text{Associative} \\ A \cap \bigg[(C' \cap B') \cup (C' \cap C)\bigg] & \text{Distributive} \\ A \cap \bigg[(C' \cap B') \cup \emptyset\bigg] & P \cap P \,=\,\emptyset \end{array}$
. . . . . . $\displaystyle \begin{array}{ccccc} A \cap (C' \cap B') &&&& P \cup \emptyset \,=\,P \\ A \cap (B' \cap C') &&&& \text{Commutative} \\ A \cap B' \cap C' &&&& \text{Associative} \end{array}$