Discrete Math Set Problem

Question

I came up a problem with for one of the question. I have to proof if the following statement is true/false.

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

I am a little confuse $C\setminus(A\setminus B)$.
I am not sure where to start.

Thanks guys.

I think it should read $C\setminus(A\setminus B)=(C\setminus A)\cup(C\cap B)$.... maybe? — Jeff Snider, Jan 20 '14 at 04:20
yeah for some reason, it don't let me the us the \ between C and (A\B) — Big Ticket, Jan 20 '14 at 04:24

score 2 · Answer 1 · answered Jan 20 '14 at 04:39

2

Using the identity $X\setminus Y=X\cap Y^C$, where $Y^C$ is the compliment of $Y$:

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

answered Jan 20 '14 at 04:39

Jeff Snider

2,817

score 0 · Answer 2 · answered Jan 20 '14 at 04:37

Here is a hint: draw a Venn diagram for three mutually intersecting sets $A$, $B$, $C$. Then shade in the region described by the LHS, and shade in the region described by the RHS. Are they the same? If so, then use the diagram to try to break down this region into smaller cases. Or, write out a truth table.

Discrete Math Set Problem

2 Answers2