Quick answer: A statement is a sentence that is either true or false, not both. Statements are joined by connectives: negation (∼p), conjunction (p ∧ q), disjunction (p ∨ q), conditional (p → q) and biconditional (p ↔ q). The conditional is false only when p is true and q is false, and it is equivalent to its contrapositive .
The revised NIMCET syllabus lists mathematical logic alongside set theory, as noted in the NIMCET revised syllabus update.
Statements
| Sentence | Statement? |
|---|---|
| "7 is a prime number." | Yes (true) |
| "2 + 2 = 5." | Yes (false) |
| "Close the door." | No, it is a command |
| "x + 2 = 5" | No, its truth depends on x |
Connectives
| Connective | Symbol | Read as | True when |
|---|---|---|---|
| Negation | not p | p is false | |
| Conjunction | p and q | both are true | |
| Disjunction | p or q | at least one is true | |
| Conditional | if p then q | except when p is true and q is false | |
| Biconditional | p if and only if q | both have the same truth value |
The basic truth table
| p | q | ∼p | p ∧ q | p ∨ q | p → q | p ↔ q |
|---|---|---|---|---|---|---|
| T | T | F | T | T | T | T |
| T | F | F | F | T | F | F |
| F | T | T | F | T | T | F |
| F | F | T | F | F | T | T |
With simple statements, a truth table has rows.
Converse, inverse and contrapositive
For the conditional :
| Name | Form | Equivalent to p → q? |
|---|---|---|
| Converse | No | |
| Inverse | No | |
| Contrapositive | Yes |
The converse and the inverse are equivalent to each other.
Tautology and contradiction
- A tautology is always true, for example .
- A contradiction is always false, for example .
- A statement that is neither is a contingency.
Key equivalences
| Law | Equivalence |
|---|---|
| Double negation | |
| De Morgan | |
| De Morgan | |
| Conditional | |
| Negation of conditional | |
| Biconditional | |
| Distributive |
These match the laws of the algebra of sets: ∧ behaves like ∩, ∨ like ∪, and ∼ like complement.
Quantifiers
| Statement | Negation |
|---|---|
| For all x, P(x) | There exists x such that not P(x) |
| There exists x such that P(x) | For all x, not P(x) |
Example: the negation of "Every student passed" is "Some student did not pass".
Worked examples
Example 1: negating a conditional
Negate "If it rains, then the match is cancelled."
Using : "It rains and the match is not cancelled."
Example 2: identify a tautology
Is a tautology?
| p | q | p ∧ q | (p ∧ q) → p |
|---|---|---|---|
| T | T | T | T |
| T | F | F | T |
| F | T | F | T |
| F | F | F | T |
All rows are T, so it is a tautology.
Example 3: contrapositive
Write the contrapositive of "If a number is divisible by 4, then it is even."
"If a number is not even, then it is not divisible by 4."
Example 4: simplify
Simplify .
By De Morgan, .
Summary table
| Idea | Key point |
|---|---|
| p → q false | Only when p is T and q is F |
| Contrapositive | Equivalent to the original conditional |
| p → q | |
| Negation of p → q | |
| Truth table rows |
Exam tips
- Memorise the single false row of ; most logic questions depend on it.
- To test for a tautology quickly, look for a row that makes the statement false.
- When negating "and" or "or", switch the connective and negate both parts.
- "For all" becomes "there exists" under negation, and vice versa.
See all topics in the NIMCET Set Theory chapter.
Common mistakes
- Treating the converse as equivalent to the original conditional.
- Negating as ; that is the inverse.
- Forgetting that "or" in logic is inclusive: is true when both are true.
- Treating open sentences like "x + 2 = 5" as statements.
Frequently Asked Questions
When is p → q false?
Only when p is true and q is false.
What is the contrapositive of p → q?
It is ∼q → ∼p, and it always has the same truth value as p → q.
What is the negation of p → q?
The negation is p ∧ ∼q.
What is a tautology?
A tautology is a compound statement that is true for every combination of truth values, such as p ∨ ∼p.
How many rows does a truth table with three statements have?
It has 2³ = 8 rows.