How many different logical operations can there be? ⇒, ∨, ∧, and ⇔ are logical operations we have studied briefly. How many different logical operations can there be?
I'm not entirely clear on what this problem wants from me. It starts by showing the 4 logical operations above, the asks how many logical operations there can be... which makes no sense to me. Any tips you may have as to how to start this problem or what it is asking for in layman's terms, are greatly appreciated!
 A: I assume we are only talking about binary logical operators. Let $\star$ be some logical operator. The truth table for $p \star q$ will have four rows. In how many ways can you then fill out this truth table?
A: For $n$ variables, there are $2^{2^n}$ possible functions.
Indeed, the domain of the function counts $2^n$ combinations of the input variables, and for every combination you can freely assign $0$ or $1$.
For two variables, there are four input combinations, $$00|01|10|11,$$ that can respectively correspond to the outputs $$0|0|0|0,0|0|0|1,0|0|1|0,0|0|1|1,0|1|0|0,0|1|0|1,0|1|1|0,0|1|1|1,\\
1|0|0|0,1|0|0|1,1|0|1|0,1|0|1|1,1|1|0|0,1|1|0|1,1|1|1|0,1|1|1|1.$$
Among these you will recognize familiar operators, like
$$\land: 00|01|10|11\to 0|0|0|1\\
\oplus: 00|01|10|11\to1|0|0|1$$
and degenerate functions, such as constant or function of a single variable
$$0:00|01|10|11\to 0|0|0|0\\
\text{left operand}:00|01|10|11\to 0|0|1|1.$$
A: I'm assuming unary and binary operations and standard boolean (2-valued) logic.
Let's start with unary operations: Those are operations that take one truth value, and return another one. There are two truth values, and for each truth value you can independently choose a result, so there are $2^2=4$ different functions.
However that includes the two constant functions $x\mapsto F$ and $x\mapsto T$; while those are of course valid functions, I would consider a logical operation only a function that actually depends on its input. By subtracting the two constant functions from the total of four functions, there remain two unary logical operations:


*

*The identity, that just returns its argument.

*The negation $\lnot$, that exchanges $T$ and $F$.


For binary operations, we note that two logical variables can have $2^2=4$ different values, for a total of $2^4=16$ logical values. However that again includes the two constant functions, and in addition functions that ignore one of their arguments. There are four of the latter, namely two that return one of their arguments, and two that return the negation of one of their arguments.
So when removing those functions which don't actually depend on both arguments, we get $16-6=10$ binary logical operations. Those can be further classified by the number of input combinations which give a true result:


*

*$3$ input combinations giving $T$: $\land$, $\implies$, $\impliedby$, $\bar\lor$ (the latter meaning NOR, that is $a\bar\lor b \iff \lnot (a\lor b)$.

*$2$ input combinations giving $T$: $\iff$, $\oplus$ (the latter being the XOR operation)

*$1$ input combination giving $T$: $\lor$, the negations of $\implies$ and $\impliedby$ (I don't know any symbols for those), $\bar\land$ (NAND).

