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I've been attempting this problem for a while now, and I'm confused as to how the final answer should look like.

The question in greater detail:

Suppose that F ↝ 0 ↝ ∅ and T ↝ 1 ↝ { ∅ }. Prove that the logical operators ‘∧’ and ‘∨’ can be defined using only set operations. Hint: use a proof by cases.

Is the solution supposed to look something along the lines of: {x| x ∈ A ∨ x ∈ B}, or just A ∨ B? I've just been struggling on how I can even "define" those logical operators using set operations.

Any help or hints will be appreciated.

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    I think a better way to say this is that the "logical operations" $\land$ and $\lor$ can be *modeled* using set operations $\cap$ and $\cup$. – 2017-02-11

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Hint

$A \lor B =$ F iff $A=$ F and $B=$ F.

But F $= \emptyset$ and T $= \{ \emptyset \}$.

Thus, we want that :

$A \lor B = \emptyset$ iff $A= \emptyset$ and $B = \emptyset$.

But $A \cup B = \{ x \mid x \in A \text { or } x \in B \}$ and $A \cup B = \emptyset$ iff $A = \emptyset$ and $B = \emptyset$.

I think that the "proof by cases" hinted amounts to computing the four cases:

$\emptyset \cup \emptyset, \emptyset \cup \{ \emptyset \}, \{ \emptyset \} \cup \emptyset, \{ \emptyset \} \cup \{ \emptyset \}$.

In the same way, $A \land B =$ T iff $A=$ T and $B=$ T.

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    Your hint definitely made it a lot clearer as to what I had to do. Thank you! – 2017-02-11
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The phrasing of the problem is indeed rather unclear. Here is what I would guess is meant by "$\wedge$ can be defined using only set operations" (the case of $\vee$ is similar). Let $f:\{F,T\}\to \{\emptyset,\{\emptyset\}\}$ be the bijection described, so that $f(F)=\emptyset$ and $f(T)=\{\emptyset\}$. Then your goal is to find some expression $p(A,B)$ in terms of set operations such that if $x,y\in\{F,T\}$, then $f(x\wedge y)=p(f(x),f(y))$. In other words, if you think of $F$ as being the same as $\emptyset$ and $T$ as being the same as $\{\emptyset\}$ (so that the function $f$ does nothing), then the expression $x\wedge y$ means the same thing as $p(x,y)$.

This still leaves the question of what is meant by "set operations". Perhaps these have been defined in the context of the problem; in any case, I would guess this refers to operations like union, intersection, and complement. So for instance, an example of the sort of thing $p(A,B)$ might be is $A\cup (B\cap A)$. Play around with simple expressions you can build in terms of $A$ and $B$ using these set operations and see if you can find one that works.