Odisha State Board CHSE Odisha Class 11 Math Notes Chapter 1 Mathematical Reasoning will enable students to study smartly.

## CHSE Odisha 11th Class Math Notes Chapter 1 Mathematical Reasoning

**Proposition: (Mathematically Acceptable)**

A proposition (or mathematically acceptable statement) is a declarative sentence that is either true or false but not both.

(1) Thus a sentence will be a statement if

- It is declarative
- It has a truth value (either true (T) or false (F).

(2) A sentence cannot be a statement if it is

(i) A question

(ii) An order

(iii) An exclamation

(iv) A wish

(v) Advice or it involves

- variable time like ‘today’, ‘tomorrow’, ‘yesterday’ etc.
- Variable place like ‘here’, ‘there’, etc.
- pronouns like ‘he’, ‘she’, ‘they’ etc.
- Relative words/adjectives / undefined words like ‘good’, ‘bad’, ‘beautiful’, ‘wise’ etc
- Variable x, y, z, u, v….etc

(3) We denote statements by same letters are p, q, r, s, etc.

Negative (~): Denial of a statement is its negation.

Axiom of negation:

For any proposition p, if p is true, then ~p (Negation of p) is false and if p is false, then ~p is true,

Truth table of Negation:

p | ~p |

T | F |

F | T |

**Logical Connectives:**

- Two statements can be combined together by using the words like or, and, if, only if, if and only if etc. These are known as logical connectives.
- A proposition in which one or more connectives appear is known as a composite or compound proposition.

**Conjunction (∧), (and):**

Axiom: A conjunction p ∧ q is true if both ‘p’ and ‘q’ are both true and false otherwise.

Truth table:

p | q | p ∧ q |

T | T | T |

T | F | F |

F | T | F |

F | F | F |

**Disjunction (∨) (or):**

Axiom: A disjunction p ∨ q is false only when both ‘p’ and ‘q’ are false and is true otherwise.

Truth table:

p | q | p ∨ q |

T | T | T |

T | F | T |

F | T | T |

F | F | F |

- Inclusive and exclusive sense of ‘OR’

→ An employee either goes on leave or attends to his duty. (Exclusive)

→ In this restaurant you can order veg or non-veg items. (Inclusive)

**Conditional (→)(if … then):**

Axiom: A conditional p → q is false only when ‘p’ is true and ‘q’ is false in all other cases it is true.

Truth table:

p | q | p → q |

T | T | T |

T | F | F |

F | T | T |

F | F | T |

**Converse, Inverse and Contrapositive:**

- Converse of p → q is q → p
- Inverse of p → q is ~p → ~q
- Contra positive of p → q is ~q → ~ p

**Biconditional (p ↔ q)(p if and only if q):**

Axiom: A biconditional p ↔ q is true if both ‘p’ and ‘q’ have same truth value and is false otherwise.

Truth table:

p | q | p ↔ q |

T | T | T |

T | F | F |

F | T | F |

F | F | T |

Equivalent statements:

Two statements ‘p’ and ‘q’ are said to be equivalent statements if they have same truth values.

**Tautology:**

A statement is a tautology if it is always true:

**Implication and double implication:**

- If a conditional p → q is a tautology then we say p implies q and we write P ⇒ q
- If a biconditional p ↔ q is a tautology and we write p ⇒ q.

**Contradiction:**

A contradiction we mean a proposition that is false for all possible assignments of truth values to its prime components.

**Logical Quantifiers:**

Logical quantifiers are the words that associate a quantity to it. There are two types of logical quantifiers.

(i) Existential (There exists)

(ii) Universal (For all, for every).

**Validity Of Statements**

A statement is said to be valid if it is true.

Techniques to check the validity of a statement:

**Validity Of Statements With ‘And’**

To prove p ∧ q is true we follow the following steps :

Step – 1: Show that ‘p’ is true.

Step – 2: Show that ‘q’ is true.

**Validity Of Statements With ‘ OR’**

To prove p ∧ q is true we have to consider the following cases :

Case – 1: By assuming p is false, prove that q is true.

Case – 2: By assuming q is false, prove that p is true.

**Validity Of Statements With ‘if … then’**

To prove if ‘p’ then ‘q’ is true we can adopt any one of the following methods.

- Method – 1 (Direct Method):

Assume ‘p’ is true and prove that ‘q’ is true (i.e. p ⇒ q) - Method – 2 (Contrapositive Method):

Assume ‘q’ is false and prove that ‘p’ is false. (i.e. ~ q ⇒ ~ p) - Method – 3 (Contradiction Method):

→ Assume that p → q is false, i.e. p is true and q is false

→ Obtain an absurd result

→ This is due to our false assumption.

→ Thus by the method of contradiction p → q is true. i.e., the statement is valid. - Method – 4 (By giving a counter-example):

To prove a statement is false we give a single example where it is false.

**Validity Of Statement With ‘if and only if’.**

To prove ‘p’ if and only if ‘q’ is true we have to follow the following steps.

Step – 1: Take ‘p’ is true and prove that ‘q’ is true.

Step – 2: Take q is true and prove that ‘p’ is true.