P is determined by three logical variables X, Y and Z. For every possible combination of X, Y and Z, the truth table gives the corresponding value of P..
Construct compound proposition p according to truth table.
For each variable x, y, z, the compound proposition p is true, that is, the combination of P= 1
For example, in our truth table, the combination of P= 1 is:
X=0,Y=0,Z=0
X=0,Y= 1,Z=0
X=0,Y= 1,Z= 1
X= 1,Y=0,Z=0
X= 1,Y=0,Z= 1
Use the variables x, y, z or their negation? x,? y,? Z as a conjunction, such as:
For X=0, Y=0, Z=0, the conjunction is:? x? ∧Y? ∧Z
For X=0, Y= 1, Z=0, the conjunction is:? x? ∧ Y? ∧Z
For X=0, Y= 1, Z= 1, the conjunction is:? x? ∧ Y? ∧ Z
For X= 1, the conjunction of Y=0 and Z=0 is: x? ∧ ? y? ∧Z
For X= 1, Y=0, Z= 1, the conjunction is: x? ∧ ? y? ∧ Z
Then extract all conjunctions, namely:
(? x? ∧Y? ∧Z) ∨? (? x? ∧ Y? ∧Z)? ∨? (? x? ∧ Y? ∧ Z)? ∨? (X? ∧Y? ∧Z)? ∨? (X? ∧Y? ∧ Z)
Regarding functional integrity:
A set of logical operators, such as the ones we used above: AND ∧, or ∨, NOT? These three operators.
All compound propositions can be derived from these three operators, so they form a set of fully functional operators.
Similar full-featured operators are:
NAND: Because "or" can be represented by "NAND": X? ∨ Y =(? x? ∧Y)
NOR: Because AND can be represented by NOR: X? ∧? Y =(? X ∨Y)
……