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APL, 22 20 bytes
0000 false ≡+
0001 p and q ∧
0010 p and not q >
0011 p ⊣
0100 not p and q <
0101 q ⊢
0110 xor ≠
0111 p or q ∨
1000 not p and not q ⍱
1001 eq =
1010 not q ...