Chapter 15 – Mathematical Reasoning (JEE)
1. Introduction to Mathematical Reasoning
Mathematical reasoning deals with **logic**, **statements**, and **valid conclusions**.
In JEE, this chapter is **easy but tricky** and mostly logic-based.
This chapter does NOT require calculation.
It requires **clear understanding of logical language**.
2. Statement
A **statement** is a sentence which is either **true or false**, but not both.
- “2 is a prime number” → Statement (True)
- “x is a real number” → Not a statement
3. Simple Statement
A statement that cannot be broken into smaller statements.
4. Compound Statement
A statement formed by combining two or more simple statements using logical connectors.
5. Logical Connectives
| Symbol | Name | Meaning |
|---|---|---|
| $\land$ | AND | True if both are true |
| $\lor$ | OR | True if at least one is true |
| $\neg$ | NOT | Negation |
| $\Rightarrow$ | Implies | If p then q |
| $\Leftrightarrow$ | If and only if | Both imply each other |
6. Negation of a Statement
If statement is $p$, its negation is $\neg p$
- p: “x is even”
- ¬p: “x is not even”
7. Compound Statements Using AND
$p \land q$ is true only when **both p and q are true**.
8. Compound Statements Using OR
$p \lor q$ is true when **at least one of p or q is true**.
In mathematics, OR is **inclusive OR**, not exclusive.
9. Conditional Statement (Implication)
$p \Rightarrow q$
“If p is true, then q must be true”
Important:
If p is false, $p \Rightarrow q$ is always TRUE.
10. Converse, Inverse, Contrapositive
| Type | Form |
|---|---|
| Converse | $q \Rightarrow p$ |
| Inverse | $\neg p \Rightarrow \neg q$ |
| Contrapositive | $\neg q \Rightarrow \neg p$ |
A statement and its **contrapositive are always logically equivalent**.
11. Biconditional Statement
$p \Leftrightarrow q$ means
$p \Rightarrow q$ AND $q \Rightarrow p$
12. Truth Values
Truth value of a statement can be:
- True (T)
- False (F)
13. Truth Table
| p | q | $p \land q$ | $p \lor q$ | $p \Rightarrow q$ |
|---|---|---|---|---|
| T | T | T | T | T |
| T | F | F | T | F |
| F | T | F | T | T |
| F | F | F | F | T |
14. Tautology
A statement which is **always true**, regardless of truth values.
$p \lor \neg p$ is a tautology
15. Contradiction
A statement which is **always false**.
$p \land \neg p$ is a contradiction
16. Valid and Invalid Statements
A statement is valid if it is **logically correct** under all circumstances.
17. Mathematical Quantifiers
| Symbol | Meaning |
|---|---|
| $\forall$ | For all |
| $\exists$ | There exists |
18. Negation of Quantified Statements
$\neg(\forall x\, p(x)) = \exists x\, \neg p(x)$
$\neg(\exists x\, p(x)) = \forall x\, \neg p(x)$
19. Common JEE Traps
- Misinterpreting “if” and “only if”
- Wrong negation of statements
- Confusing OR with exclusive OR
- Ignoring contrapositive equivalence
20. Final Revision Checklist
You have mastered this chapter if you can:
- Identify statements correctly
- Write negations accurately
- Construct truth tables
- Recognize tautology and contradiction
- Handle implication and biconditional statements