# Thread: Set logic proof question

1. ## Set logic proof question

Prove: if F = a U b and a intersect b = disjoint, then a = F\b.

sorry I didnt know the latex for some of these things.

doesnt this mean that a is a subset of F and not in b?

i think i know how to prove that a is a subset of F\b but how do I prove that F\b is a subset of a?

2. Originally Posted by p00ndawg
Prove: if F = a U b and a intersect b = disjoint, then a = F\b.
sorry I didnt know the latex for some of these things.
doesnt this mean that a is a subset of F and not in b?
i think i know how to prove that a is a subset of F\b but how do I prove that F\b is a subset of a?
$$F=A\cup B$$ gives $\displaystyle F=A \cup B$
$$A\cap B= \emptyset$$ gives $\displaystyle A\cap B= \emptyset$.
$$A = F\setminus B$$ gives $\displaystyle A = F\setminus B$.

Proof.
If $\displaystyle x\in A$ then because $\displaystyle F=A\cup B ~\&~ A\cap B= \emptyset$ then what?

3. Originally Posted by Plato
$$F=A\cup B$$ gives $\displaystyle F=A \cup B$
$$A\cap B= \emptyset$$ gives $\displaystyle A\cap B= \emptyset$.
$$A = F\setminus B$$ gives $\displaystyle A = F\setminus B$.

Proof.
If $\displaystyle x\in A$ then because $\displaystyle F=A\cup B ~\&~ A\cap B= \emptyset$ then what?

A is a subset of $\displaystyle F\setminus B$?

4. Originally Posted by p00ndawg
A is a subset of $\displaystyle F\setminus B$? YES
If $\displaystyle x\in F\setminus B$ then does $\displaystyle x\in A$? WHY?
What does that tell us?

5. Originally Posted by Plato
If $\displaystyle x\in F\setminus B$ then does $\displaystyle x\in A$? WHY?
What does that tell us?
X has to be in F because it is not in B, and since A is a subset of F\B we know that A has to be in F.

that right?

6. Originally Posted by p00ndawg
X has to be in F because it is not in B, and since A is a subset of F\B we know that A has to be in F. that right?
YES