UNIT 4 – Truth Table Solving
Truth table means checking all possible True (T) and False (F) values.
For two propositions P and Q, possible combinations are:
| P | Q |
|---|---|
| T | T |
| T | F |
| F | T |
| F | F |
These four combinations are always used.
1. Truth Table of Negation (¬P)
Definition
Negation changes truth value to opposite.
If P is true → ¬P false If P false → ¬P true
Table
| P | ¬P |
|---|---|
| T | F |
| F | T |
Example
P = Today is Sunday ¬P = Today is not Sunday
2. Truth Table of Conjunction (AND)
Symbol = P ∧ Q
Definition
Conjunction is true only when both statements are true.
Table
| P | Q | P∧Q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
Example
P = I study Q = I pass
P ∧ Q = I study AND I pass
Both must happen.
3. Truth Table of Disjunction (OR)
Symbol = P ∨ Q
Definition
Disjunction is true if at least one is true.
Table
| P | Q | P∨Q |
|---|---|---|
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
Example
P = I eat rice Q = I eat bread
P ∨ Q = I eat rice OR bread
One true is enough.
4. Truth Table of Implication
Symbol = P → Q
Definition
If P, then Q.
Table
| P | Q | P→Q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
Important
Only T → F is false.
Example
P = I study Q = I pass
If I study, then I pass.
5. Truth Table of Bi-conditional
Symbol = P ↔ Q
Definition
“If and only if.”
True when both are same.
Table
| P | Q | P↔Q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
Example
P = Switch ON Q = Bulb ON
Same condition.
6. Combined Truth Table
This is highly asked.
| P | Q | P∧Q | P∨Q | P→Q | P↔Q |
|---|---|---|---|---|---|
| T | T | T | T | T | T |
| T | F | F | T | F | F |
| F | T | F | T | T | F |
| F | F | F | F | T | T |
Memorize.
Tautology Check Using Truth Table
Definition
If every output is true → tautology.
Example:
P ∨ ¬P
| P | ¬P | P∨¬P |
|---|---|---|
| T | F | T |
| F | T | T |
All true → tautology.
Contradiction Check
Definition
If every output false → contradiction.
Example:
P ∧ ¬P
| P | ¬P | P∧¬P |
|---|---|---|
| T | F | F |
| F | T | F |
All false → contradiction.
Important Short Answer Questions
1. Define statement.
A statement is a sentence that has a definite truth value, either true or false.
2. Define proposition.
A proposition is a declarative statement that is either true or false.
3. Define negation.
Negation means opposite of a statement.
4. Define conjunction.
Conjunction is a logical operation formed by joining two statements using AND.
5. Define disjunction.
Disjunction is a logical operation formed by joining two statements using OR.
6. Define implication.
Implication is a logical statement of the form If P, then Q.
7. Define tautology.
A tautology is a statement that is always true.
8. Define contradiction.
A contradiction is a statement that is always false.
9. State De Morgan’s Laws.
- NOT (P AND Q) = NOT P OR NOT Q
- NOT (P OR Q) = NOT P AND NOT Q
Important Long Questions
Prepare these:
- Explain proposition with examples.
- Explain conjunction, disjunction and implication with truth table.
- Explain converse, inverse and contrapositive.
- Explain tautology and contradiction.
- Write De Morgan’s Laws with examples.
- Explain logical equivalence.