Download Logic Summary: Symbols, formulas, truth tables A truth table

Logic Summary: Symbols, formulas, truth tables
A truth table involving n different statements will have 2n rows.
Examples: The truth table for (~p V q) V r has 23=8 rows and the truth table for p V (q → ~q) has 22 rows.
Connective
and
or
not
If …then
If and only if
Symbol
Λ
V
~
→
↔
Name
Conjunction
Disjunction
Negation
Implication
Biconditional
The 5 BASIC truth tables:
Negation, 'not p'
p
~p
T
F
F
T
Conjunction, 'p and q'
"T to F, F to T."
Disjunction, 'p or q'
p
q
pΛq
p
q
pVq
T
T
F
F
T
F
T
F
T
F
F
F
T
T
F
F
T
F
T
F
T
T
T
F
"Only True if both are True."
"Only False if both are False."
Implication, 'If p, then q'
Biconditional, 'p if and only if q'
p
q
p→q
p
q
p↔q
T
T
F
F
T
F
T
F
T
F
T
T
T
T
F
F
T
F
T
F
T
F
F
T
"Only False if p is True and q is False."
DeMorgan’s Laws
~(p Λ q) ≡ ~p V ~ q
~(p V q) ≡ ~p Λ ~ q
"Only True if p and q have the same truth value."
Statement: p→q (If p, then q): If you go, then I stay.
Converse: q→p (If q, then p): If I stay, then you go.
Inverse: ~p→~q (If not p, then not q): If you do not go, then I will not stay.
Contrapositive: ~q→~p (If not q, then not p): If I do not stay, then you do not go.
A statement p → q is equivalent to its contrapositive ~q → ~p.

For p → q, p is the ‘hypothesis’ and q is the ‘conclusion’.

Quantifiers: Universal (All, Each, Every, No, None) and Existential (Some, There exists, At least one)
NEGATION OF A STATEMENT WITH A UNIVERSAL QUANTIFIER WILL INVOVLE AN EXISTENTIAL QUANTIFIER.
NEGATION OF A STATEMENT WITH AN EXISTENTIAL QUANTIFIER WILL INVOVLE A UNIVERSAL QUANTIFIER.
(Some tips on negating statements with quantifiers are on the following page.)

Two statements are logically equivalent if their truth tables are identical
Basic circuit
(with a gate that can open and close)
Circuit "in series" (Gates lined up one after the other)
-'Current' will only flow all the way across if both gates are closed.
-If 'open' is False & 'closed' is True, this circuit is logically equivalent
to the CONJUNCTION ( p Λ q ).
Circuit "in parallel" (Gates are arranged one above the other)
-'Current' will not flow all the way across only if both gates are open.
-If we define 'open' as False & 'closed' as True, this circuit is
logically equivalent to the DISJUNCTION ( p V q ).
Here are a few tips for negating statement with quantifiers.
The negation of an "All do/are" statement
["All athletes are wealthy."]
is a "Some do not/are not" statement
["Some athletes are not wealthy."]
or a "There exists at least one not" statement
["There exists at least one athlete who is not wealthy."]
The negation of an "All do not/are not" statement
["It is true for all athletes that they are not wealthy."]
is a "Some do/are" statement
["Some athletes are wealthy."]
or a "There exists at least one" statement
["There exists at least one athlete who is wealthy."]
The negation of a "Some do/are" statement
["Some students get scholarships."]
is a "No/none do" statement
["No student gets a scholarship."]
or a "All do not/are not" statement
["It is true for all students that they do not get scholarships."]
The negation of a "Some do not/are not" statement
["Some students do not get scholarships."]
is a "No/none do not" statement
["No student does not get a scholarship."]
or a "All do/are" statement
["All students get scholarships."]