What is $(x ∈ S) ∧(x∉S) $ mean? I am just wondering what does $(x  ∈ S) ∧ (x ∉ S)$ mean. Anyone can explain it to me?
I am in the midst of doing my homework, and I get something like this:
$((x∈S)∧(x∉S)) ∨ ((x∈S)∧(x∉T))$. I know that the answer must be $(x∈(S-T))$
But whats the reasoning? Is it meaningless so that we can omit it?
 A: The statement $(x\in S)$∧$(x\not\in S)$ means: ($x$ is contained in $S$) AND ($x$ is not contained in $S$)
Which is logically false, since $x$ cannot both be in $S$ and not in $S$ at the same time.
The statement $(x\in S)$∧$(x\not\in T)$ means:  ($x$ is contained in $S$) AND ($x$ is not contained in $T$)
Going back to the full statement: $[(x\in S)$∧$(x\not\in S)]$∨$[(x\in S)$∧$(x\not\in T)]$
We have:  [($x$ is contained in $S$) AND ($x$ is not contained in $S$)] OR [($x$ is contained in $S$) AND ($x$ is not contained in $T$)]
Since the left hand side is always false, the OR condition will be true only if the right hand side is true, so we can disregard the left hand side, giving us: 
($x$ is contained in $S$) AND ($x$ is not contained in $T$)
Another way of writing that is: ($x$ is contained in $S$ but not $T$)  i.e. $x\in S-T$
A: When you get something like $((x \in S) \wedge (x \not\in S)) \vee ((x \in S) \wedge (x \not\in T))$ you can "translate it" to $ x \in (S \cap S^c) \cup ( S \cap T^c)$. Now it is clear that $ S \cap S^c = \emptyset$. So $(S \cap S^c) \cup ( S \cap T^c) = \emptyset \cup (S \cap T^c) = S \cap T^c$. Therefore $x \in S \cap T^c \rightarrow x \in S \backslash T$.
