0,00,11,01,1(p,q)
0 0 0 0 false
0 0 0 1 AND
0 0 1 0 https://en.wikipedia.org/wiki/Material_nonimplication
0 0 1 1 p
0 1 0 0 https://en.wikipedia.org/wiki/Converse_nonimplication
0 1 0 1 q
0 1 1 0 XOR, !=
0 1 1 1 OR
1 0 0 0 NOR
1 0 0 1 XNOR, ==
1 0 1 0 !q
1 0 1 1 https://en.wikipedia.org/wiki/Converse_implication
1 1 0 0 !p
1 1 0 1 https://en.wikipedia.org/wiki/Material_conditional
1 1 1 0 NAND
1 1 1 1 true
⤫ 
non-commutative
⤭ 
commutative

Note: Based on https://en.wikipedia.org/wiki/Truth_table#Binary_operations BUT with reordering the rows of the first table for our sort order