Symbolic logic operator "implication" to logical gates
A simple solution for such puzzles is counting the one and zeroes at the output:


all zero or all one → trivial
one zero and three ones → OR gate (or NAND gate)
one one and three zeroes → AND gate (or NOR gate)
two zeroes and two ones → trival (copy P or Q) or XOR or XNOR gate




P Q   R
1 1   1
1 0   0
0 1   1
0 0   1


Three ones. So we have an OR gate at the output. Let's take a look at the case where R is zero: An OR gate needs two zeroes to create a zero. So we have to invert P before feeding it into the gate.

P  -P Q   R
1   0 1   1
1   0 0   0
0   1 1   1
0   1 0   1


Let's order the lines the "normal" way:

P  -P Q   R
1   0 0   0
1   0 1   1
0   1 0   1
0   1 1   1


I think you can see now the solution is inverting the P input, then feeding -P and Q into an OR gate.