“8 is an even number.”
This statement is true.
🔀 Logic and Truth Tables
Learn how statements become true or false, combine them using logical operators and evaluate conditions used in mathematics, algorithms and computer programs.
🎯 What You Will Learn
01
Identify propositions and determine their truth values.
02
Use NOT, AND, OR, XOR, implication and biconditional.
03
Construct and interpret complete truth tables.
04
Translate logical expressions into programming conditions.
💬 1. Statements and Propositions
A proposition is a declarative statement that has exactly one truth value: either true or false.
“15 is divisible by 4.”
This statement is false.
“Close the window.”
A command does not have a truth value.
“What is your age?”
A question is neither true nor false.
Quick check:
If you can meaningfully label a sentence as true or false,
it is usually a proposition.
🔁 2. Truth Values and Negation
We write T for true and F for false. The NOT operator reverses a truth value.
NOT P = ¬P
| P | ¬P |
|---|---|
| T | F |
| F | T |
Example
P: “A student submitted the form.”
¬P: “A student did not submit the form.”
¬P: “A student did not submit the form.”
🔗 3. AND, OR and XOR Operators
Both must be true
The result is true only when P and Q are both true.
At least one is true
The result is false only when both statements are false.
Exactly one is true
The result is true when P and Q have different values.
| P | Q | P ∧ Q | P ∨ Q | P ⊕ Q |
|---|---|---|---|---|
| T | T | T | T | F |
| T | F | F | T | T |
| F | T | F | T | T |
| F | F | F | F | F |
In logic, OR normally means inclusive OR:
one or both statements may be true. XOR is used when exactly
one must be true.
➡️ 4. Implication and Biconditional
If P, then Q
It is false only when P is true but Q is false.
P if and only if Q
It is true when P and Q have the same truth value.
| P | Q | P → Q | P ↔ Q |
|---|---|---|---|
| T | T | T | T |
| T | F | F | F |
| F | T | T | F |
| F | F | T | T |
Memory rule:
A promise “If P, then Q” is broken only when P happens but Q
does not happen.
🧱 5. How to Build a Truth Table
A truth table checks an expression under every possible combination of truth values.
- Count the variables. For example, P and Q means two variables.
- Calculate the rows. Use 2n, where n is the number of variables.
- List all combinations. For two variables: TT, TF, FT and FF.
- Create intermediate columns. Evaluate NOT operations before larger expressions.
- Evaluate the final column. Apply operators according to precedence.
1 variable2¹ = 2 rows
2 variables2² = 4 rows
3 variables2³ = 8 rows
4 variables2⁴ = 16 rows
🧮 6. Compound Expressions and Precedence
A compound expression contains more than one operator. Use parentheses whenever they make the intended order clearer.
Parentheses → NOT → AND → OR → Implication → Biconditional
Worked Example
Evaluate (P ∨ Q) ∧ ¬P when P = F and Q = T.
Solution
First, P ∨ Q = F ∨ T = T.
Next, ¬P = ¬F = T.
Finally, T ∧ T = T.
Final truth value: T
⚖️ 7. Equivalence and De Morgan’s Laws
Two expressions are logically equivalent when their final truth-table columns match for every row.
“If P then Q” can be rewritten using NOT and OR.
NOT of AND becomes OR of the negated statements.
NOT of OR becomes AND of the negated statements.
Tautology
P ∨ ¬P
Contradiction
P ∧ ¬P
Contingency
P ∧ Q
🧪 Truth Table Builder
Select a logical expression. The builder evaluates it for all four combinations of P and Q and explains its meaning.
| P | Q | Result |
|---|
💻 8. Logic in Programming
Programming conditions use the same reasoning as mathematical logic. The symbols may change, but the truth rules remain the same.
| Logic | C / Java-style symbol | Example meaning |
|---|---|---|
| NOT | ! |
!isEmpty: the collection is not empty |
| AND | && |
age >= 18 && hasId: both requirements hold |
| OR | || |
isAdmin || isOwner: either permission is enough |
| XOR idea | != for Boolean values |
Exactly one of two Boolean choices is true |
CodeBhavya Example
A student may enter the lab when the booking is valid AND
the identity card is active:
validBooking && activeId.
🧩 9. Solved Problems
Problem 1
Evaluate P ∧ ¬Q when P = T and Q = F.
Solution
¬Q = ¬F = T.
P ∧ ¬Q = T ∧ T = T.
Answer: T
Problem 2
Classify P ∨ ¬P.
Solution
If P is true, P ∨ ¬P is T ∨ F = T.
If P is false, P ∨ ¬P is F ∨ T = T.
Answer: It is a tautology.
Problem 3
A portal permits access when a user is an administrator OR has both a valid account and an active subscription. Write the logical expression.
Solution
Let A = administrator, V = valid account and S = active subscription.
The two valid paths are A, or V together with S.
Answer: A ∨ (V ∧ S)
✍️ 10. Practice Problems
Decide each answer before opening the solution. Use a small truth table whenever the result is not immediately clear.
1. Is “x + 4 = 10” a proposition when x has not been specified?
No. Its truth value depends on the unknown value of x, so it
is an open sentence rather than a proposition.
2. Evaluate P ∨ Q when P = F and Q = T.
OR is true when at least one statement is true.
Therefore, F ∨ T = T.
3. When is P → Q false?
An implication is false only when P is true and Q is false.
4. Apply De Morgan’s law to ¬(P ∨ Q).
Negate both statements and change OR to AND:
¬P ∧ ¬Q.
5. Evaluate (P ⊕ Q) ∧ R for P = T, Q = F and R = T.
T ⊕ F = T because exactly one is true. Then T ∧ T =
T.
📝 Quick Revision
- A proposition has exactly one truth value: true or false.
- NOT reverses a truth value.
- AND needs both inputs to be true.
- OR needs at least one input to be true.
- XOR needs exactly one input to be true.
- P → Q is false only for P = T and Q = F.
- P ↔ Q is true when P and Q match.
- n variables require 2n truth-table rows.
- Equivalent expressions have identical final columns.
- Programming conditions use the same Boolean truth rules.
INTERACTIVE PRACTICE
Start 20-Question Test →
🔀 Logic and Truth Tables Practice
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