An Introduction to Logical Connectives and Truth Table Construction
What is a Truth Table?
A truth table is a mathematical table used in logic to determine
the truth value of a compound statement based on the truth values of its components.
Each row represents a possible combination of truth values for the variables involved.
The two possible truth values are True (T) and False (F).
For a statement with n variables, there are 2n possible rows.
Example: The statement p ∧ q ("p AND q") is only True
when both p and q are True.
Common Logical Connectives
Symbol
Name
Read As
True When…
¬p
Negation
"NOT p"
p is False
p ∧ q
Conjunction
"p AND q"
Both p and q are True
p ∨ q
Disjunction
"p OR q"
At least one of p, q is True
p → q
Conditional
"If p, then q"
p is False, OR q is True
p ↔ q
Biconditional
"p if and only if q"
p and q have the same value
p ⊕ q
Exclusive OR
"p XOR q"
Exactly one of p, q is True
Basic Truth Tables
Negation (¬p)
p
¬p
T
F
F
T
Conjunction (p ∧ q)
p
q
p ∧ q
T
T
T
T
F
F
F
T
F
F
F
F
Disjunction (p ∨ q)
p
q
p ∨ q
T
T
T
T
F
T
F
T
T
F
F
F
Conditional (p → q)
p
q
p → q
T
T
T
T
F
F
F
T
T
F
F
T
Biconditional (p ↔ q)
p
q
p ↔ q
T
T
T
T
F
F
F
T
F
F
F
T
Compound Statements
Complex logical expressions can be broken down column by column in a truth table.
Evaluate inner expressions first, then use their results to compute the outer expression.
Example: ¬p ∨ q
p
q
¬p
¬p ∨ q
T
T
F
T
T
F
F
F
F
T
T
T
F
F
T
T
Notice that ¬p ∨ q has the same truth values as p → q —
they are logically equivalent!
Tautologies & Contradictions
A tautology is a statement that is always True regardless of the
truth values of its variables. A contradiction is always False.
Tautology: p ∨ ¬p
p
¬p
p ∨ ¬p
T
F
T
F
T
T
Contradiction: p ∧ ¬p
p
¬p
p ∧ ¬p
T
F
F
F
T
F
Truth table calculator
Variables: single uppercase letters (A–Z) · Operators: AND && & · OR || | · NOT ! ~ · XOR ^ · NAND · XNOR · NOR · → (implies) · ↔ (biconditional)
Insert:
Enter an expression above to generate a truth table
Practice Problems
Problem 1:
What is the truth value of ¬p when p = True?
False. Negation flips the truth value, so ¬T = F.
Problem 2:
Evaluate p ∧ q when p = True and q = False.
False. A conjunction is True only when both operands are True.
Since q = F, the result is F.
Problem 3:
Evaluate p → q when p = False and q = False.
True. A conditional is only False when the hypothesis (p) is True and the
conclusion (q) is False. Since p = F here, the whole statement is True (vacuously true).
Problem 4:
Build the full truth table for p ↔ q.
p
q
p ↔ q
T
T
T
T
F
F
F
T
F
F
F
T
Problem 5:
How many rows does a truth table with 3 variables (p, q, r) have?
8 rows. With n variables there are 2n combinations.
23 = 8.
Challenge Problems
Construct a full truth table for the compound statement
(p ∧ q) → r with three variables p, q, and r.
Show that p → q and ¬p ∨ q
are logically equivalent by building both columns in a single truth table.
Is (p ∨ q) ∧ ¬(p ∧ q) a tautology, contradiction, or neither?
Build the truth table to justify your answer. (Hint: this is XOR!)