Mathematical Reasoning ISC Class-11 ML Aggarwal Solutions Chapter-3. Step by step Solutions of ML Aggarwal ISC Class-11 Mathematics with Exe-3.1, Exe-3.2, Exe-3.3, Exe-3.4, Exe-3.5, Exe-3.6, Exe-3.7, and Chapter Test Questions. Visit official Website CISCE for detail information about ISC Board Class-11 Mathematics.

## Mathematical Reasoning ISC Class-11 ML Aggarwal Solutions Chapter-3

Board | ISC |

Class | 11 |

Subject | Mathematics |

Chapter- | Mathematical Reasoning |

Session | 2024-25 |

Topics | Solutions of ML Aggarwal |

**Mathematical Reasoning**

**Simple Statements : **A statement is called simple if it cannot be broken down into two or more statements.

**Compound Statements : **A compound statement is the one which is made up of two or more simple statement.

**Connectives : **The words which combine or change simple statements to form new statements or compound statements are called connectives.

**Conjunction : **If two simple statements p and q are connected by the word ‘and’, then the resulting compound statement “p and q” is called a conjunction of p and q is written in symbolic form as “p ∧ q”.

Note:

- The statement p ∧ q has the truth value T (true) whenever both p and q have the truth value T.
- The statement p ∧ q has the truth value F (false) whenever either p or q or both have the truth value F.

**Disjunction :**

If two simple statements p and q are connected by the word ‘or’, then the resulting compound statement “p or q” is called disjunction of p and q and is written in symbolic form as “p ∨ q”.

Note:

- The statement p ∨ q has the truth value F whenever both p and q have the truth value F.
- The statement p ∨ q has the truth value T whenever either p or q or both have the truth value T.

Negation: An assertion that a statement fails or denial of a statement is called the negation of the statement. The negation of a statement p in symbolic form is written as “~p”.

Note:

- ~p has truth value T whenever p has truth value F.
- ~p has truth value F whenever p has truth value T.

**Negation of Conjunction:**

The negation of a conjunction p ∧ q is the disjunction of the negation of p and the negation of q.

Equivalently we write ~ (p ∧ q) = ~p ∨ ~q.

**Negation of Disjunction : **The negation of a disjunction p v q is the conjunction of negation of p and the negation of q. Equivalently, we write ~(p ∨ q) = ~p ∧ ~q.

**Negation of Negation : **Negation of negation of a statement is the statement itself. Equivalently, we write ~(~p) = p

**The Conditional Statement :**

If p and q are any two statements, then the compound statement “if p then g” formed by joining p and q by a connective ‘if-then’ is called a conditional statement or an implication and is written in symbolically p → q or p ⇒ q, here p is called hypothesis (or antecedent) and q is called conclusion (or consequent) of the conditional statement (p ⇒ q).

#### Quantifiers

Quantifiers are phrases like “There exists” and “For all”. A word closely connected with “there exists” is “for every” (or for all). Hence, the words “And” and “Or” are called connectives and “There exists” and “For all” are called quantifiers.

#### Implications in Mathematical Reasoning

It is possible to write statements with implications such as “If ”, “only if ”, “ if and only if ”. The statements with “if-then” are very common in mathematics.

A sentence with if p, then q can be written in the following ways.

- p implies q (denoted by p ⇒ q)
- p is a sufficient condition for q
- q is a necessary condition for p
- p only if q
- ∼q implies ∼p

‘If and only if’, represented by the symbol ‘⇔‘ means the following equivalent forms for the given statements p and q.

(i) p if and only if q

(ii) q if and only if p

(iii) p is necessary and sufficient condition for q and vice-versa

(iv) p ⇔ q

#### Contrapositive and converse

Contrapositive and converse are certain other statements which can be formed from a given statement with “if-then”.

- The contrapositive of a statement p ⇒ q is the statement ∼ q ⇒ ∼p .
- The converse of a statement p ⇒ q is the statement q ⇒ p.
- p ⇒ q together with its converse, gives p if and only if q

#### Validating Statements

The following methods are used to check the validity of statements:

(i) direct method

(ii) contrapositive method

(iii) method of contradiction

(iv) using a counter example

**Exe-3.1**

Mathematical Reasoning ISC Class-11 ML Aggarwal Solutions Chapter-3

**Exe-3.2**

Mathematical Reasoning ISC Class-11 Solutions Chapter-3

**Exe-3.3**

Mathematical Reasoning ISC Class-1 Solutions Chapter-3

**Exe-3.4**

Mathematical Reasoning ISC Class-11 ML Aggarwal Solutions Chapter-3

**Exe-3.5**

Mathematical Reasoning ISC Class-11 ML Aggarwal Solutions Chapter-3

**Exe-3.6**

Mathematical Reasoning ISC Class-11 ML Aggarwal Solutions Chapter-3

**Exe-3.7**

Ch-3 Mathematical Reasoning ISC Class-11 Mathematics

**Chapter Test**

Mathematical Reasoning ISC Class-11 ML Aggarwal Solutions Chapter-3

**-: End of Mathematical Reasoning ISC Class-11 ML Aggarwal Solutions Chapter-3 :-**

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