Question

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

Answer 1

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.