The Truth Tables between two variables, A and B. There are 16 combinations. I'm comparing the logic notation, the set notation, the Venn diagram, and the truth table. I'm also pairing each with its complement. In some cases the complement notation can be simplified by DeMorgan's Laws:
¬(P∧Q)=¬P∨¬Q
¬(P∨Q)=¬P∧¬Q

 

A and B
A∧B
      
logic
  
¬(A∧B)
A∩B
set
  
(A∩B)′
A
B
AND(A,B)
T
T
T
T
F
F
F
T
F
F
F
F
  A and B
A
B
NAND(A,B)
T
T
F
T
F
T
F
T
T
F
F
T
 
A or B
A∨B
logic
  
¬(A∨B)
A∪B
set
  
(A∪B)′
A
B
OR(A,B)
T
T
T
T
F
T
F
T
T
F
F
F
  A or B
A
B
NOR(A,B)
T
T
F
T
F
F
F
T
F
F
F
T
 
A xor B
(A∧¬B)∨(¬A∧B)
logic
  
A↔B
(A∪B)−(A∩B)
set
  
((A∪B)−(A∩B))′
A
B
XOR(A,B)
T
T
F
T
F
T
F
T
T
F
F
F
  A xor B
A
B
XNOR(A,B)
T
T
T
T
F
F
F
T
F
F
F
T
 
implication (A implies B)
A→B
logic
  
¬(A→B)
A′∪B
set
  
A∩B′
A
B
A→B
T
T
T
T
F
F
F
T
T
F
F
T
  A implies B
A
B
¬(A→B)
T
T
F
T
F
T
F
T
F
F
F
F
       
       
always true
1
logic
  
0
U
set
  
∅
A
B
TRUTH(A,B)
T
T
T
T
F
T
F
T
T
F
F
T
  universal
A
B
FALSE(A,B)
T
T
F
T
F
F
F
T
F
F
F
F
       
       
always A
A
logic
  
¬A
A
set
  
A′
A
B
A
T
T
T
T
F
T
F
T
F
F
F
F
  A
A
B
NOT(A)
T
T
F
T
F
F
F
T
T
F
F
T
       
       
always B
B
logic
  
¬B
B
set
  
B′
A
B
B
T
T
T
T
F
F
F
T
T
F
F
F
  B
A
B
NOT(B)
T
T
F
T
F
T
F
T
F
F
F
T
       
       
implication (B implies A)
B→A
logic
  
¬(B→A)
A∪B′
set
  
A′∩B
A
B
B→A
T
T
T
T
F
T
F
T
F
F
F
T
  B implies A
A
B
¬(B→A)
T
T
F
T
F
F
F
T
T
F
F
F