Justify and show your work for full credit.

Explicitly check that all hypotheses are satisfied when using theorems. Improper use of $($or the absence$)$ of proper notation will be penalized.

$(6$ pts$)$ The big exciting reveal from the first week's lectures is that we can rigorously define binary logical operations using truth tables. Note, a "rigorous definition" in this context means that it is defined in a way that avoids ambiguity. A truth table gets around the issue of how different people might interpret the words "and" and $"$or$",$ by literally defining the output of each of these statements in a table.

$($a$)(5$ pts$)$ So far we have defined the binary logical operations, AND${(}^{{?}^{?}}),OR(vv),$ XOR $(o+),$ and IF$-$THEN $().$ For your first exercise in this first real homework, I would like you to define your own binary logical operation $($different from the $4$ just listed$)$ using a truth table, give it a symbol, name it$,$ and briefly describe what it does. For example a description for the operator AND could be "AND returns true only when both the inputs statements are true and false otherwise."

$($b$)(1$ pt$)$ With AND, OR$,$ XOR, IF$-$THEN, and your new operation we have defined $5$ distinct operations. How many distinct logical binary operations are there? Just $5?6,?$ Somewhere in$-$between? Here I am not looking for a proof, but we need to see an explanation of how you came upon your answer $($Note binary operations are the same if for every input they give the same output and they are distinct if for some input they give different outputs$).$