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.