simple logic to use only one operator to construct a function

Question

So I was doing some of the problems on the book for discrete mathematics and I encountered this problem:

We define a new operator ⊙ as follows:

x ...y.... x⊙y

F... F... T

F... T... F

T... F... F

T... T... F

Using ONLY this operator construct functions that evaluate to the following boolean functions:

f(p,q) = p ^ q

Can anyone explain step by step or from the scratch how to tackle this one? I am new to this logic mathematics..

All I can think of is x⊙y = ~(p∨q)

Answer 1

You're right that $x\odot y$ is equivalent to $\neg (x\vee y)$ (I assume the switch from $x$ and $y$ to $p$ and $q$ was a typo), but this isn't the most natural way of attacking it: it would be better to view $x\odot y$ as $$(\neg x)\wedge (\neg y).$$ From here it should be clear that, if we could only build negation out of $\odot$, we'd know how to build $\wedge$ out of $\odot$. So: using $\odot$, can you express "$\neg x$"?