Question

1. Suppose that A , B and C are sets. Show that A \ (B U C) (A \ B) n (A \ C).

Answer 1

Step-by-step explanation:

We want to show that

$=A \setminus (B \cup C) = (A\setminus B) \cap (A\setminus C)$

To prove it we just use the definition of $X\setminus Y = X \cap Y^c$

So, we start from the left hand side:

$=A \setminus (B \cup C) = A \cap (B \cup C)^c$ (by definition)

$=A \cap (B^c \cap C^c)$ (by DeMorgan's laws)

$=A \cap B^c \cap C^c$ (since intersection is associative)

$=A \cap B^c \cap A \cap C^c$ (since intersecting once or twice A doesn't make any difference)

$=(A \cap B^c) \cap (A \cap C^c)$ (since again intersection is associative)

$=(A\setminus B) \cap (A \setminus C)$ (by definition)

And so we have reached our right hand side.