# Math Help - Boolean algebra / venn diagrams

1. ## Boolean algebra / venn diagrams

For any two subsets A and B of some set S,

$(A \cup B) - (A \cap B) = (A - B) \cup (B - A)$

where $X-Y = X \cap Y'$

Show that this is true by using a) Venn diagrams and b) algebraic manipulation using the properties of Boolean algebra.

Ok so im struggling to start on this. Part a i don't know how to start really. Part b I'm tempted to write

$X - Y = (X - Y')\cup(Y' - X)$

but doubt its right :/

2. Draw two circles which intersect in a Venn diagram, label one $A$ and the other $B$. Figure out the corresponding regions represented by the each expression an it should become clear.

For b) use distributivity and De Morgan's:

$(A \cup B) \cap (A \cap B)^c=(A \cup B) \cap (A^c \cup B^c)$ $
= [(A \cup B) \cap A^c] \cup [(A \cup B) \cap B^c]
= (B \cap A^c) \cup (A \cap B^c)$