By going through these CBSE Class 11 Maths Notes Chapter 14 Mathematical Reasoning Class 11 Notes, students can recall all the concepts quickly.
Mathematical Reasoning Notes Class 11 Maths Chapter 14
Statement : A sentence is called mathematically acceptable statement, if it is either true of false but not both.
Negation of a Statement: The denial of a statement is called negation of the statement e.g.
If p : Diagonals of a rectangle are equal.
So, ~ p : Diagonals of a rectangle are not equal. This may also be written as
~ p : It is false that diagonals of a rectangle are equal. Further, it may also be written as
~ p : There is at least one rectangle whose diagonals are not equal.
Compound Statement: A compound statement is a statement which is made up of two or more statements.
Each statement is called a component statement.
The Connecting Word “And” : We can connect the two statements by the word “AND”, e.g. ‘
p : 48 is divisible by 4.
q : 48 is divisible by 6.
p and q : 48 is divisible by 4 and 6.
Truth Value of p and q : The statementp and q are both true, otherwise it is false, i.e., it is false, when
(i) p is true and q is false.
(ii) p is false and q is true.
(iii) p is false and q is false.
The Connecting Word “OR” : The statement p and q may be connected by the connecting word ‘OR’: i.e., p or q, e.g.
p : Ice cream is available at dinner.
q : Coffee is available at dinner.
p or q : Ice cream or coffee is available at dinner.
Exclusive “OR” : In a statement p or q, if one of the statements either p or q is true, then the statement is true. Thus, connecting of word “OR” is exclusive.
Inclusive “OR” : In a statement either both are true, then connecting word ‘OR’ is inclusive, e.g.
At plus one level, a student can either opt for Biology or Mathematics or both.
Here, the connecting word “OR” is inclusive.
Truth value of p or q : When p and q statements both are false, then p or q is also false, otherwise it is true.
Thus, p or q is true, when
(i) p is true, q is false.
(ii) p is false, q is true.
(iii.) p and q both are true.
Quantifier “There Exists” : There exists, is used for at least one e.g.
p : There exist, a quadrilateral whose all sides are equal.
The statement is equivalent to
There is at least one quadrilateral whose all sides are equal.
Quantifier “For All” : In mathematical statements “for all” is commonly used e.g. For all n ∈ N implies that each n is a natural rumber.
Implications : “If then”, “only if’ and “if and only if’ are known as imphcations. .
If p then q : The statement if “p then q” says that in the event if p is true, the i q must be true e.g.
‘ I a numbei ” a multiple of 4, then it is a multiple of 2.
Here p : A ni her is a multiple of 4.
and q : The m. aber is a multiple of 2.
When p is true i.e., a number is a multiple of 4, then q is true
i. e., the number is a multiple of 2.
“If p then q” may be used as : l.p implies q : It is denoted by p ⇒ q. The symbol ⇒ stands for implies.
2. p is sufficient condition for q : A number is a multiple of 4 is sufficient to conclude that the number is a multiple of
3. p only if q. The number is a multiple of 4 only if it is a multiple of 2.
4. q is a necessary condition for p. When the number is a multiple of 4, it is necessarily a multiple of 2.
5. – q implies ~ p (~ stands for negation or not)
Converse Statement : The converse of “if p then q” is if q then p e.g.
If a number x is odd, then x2 is also odd.
Its converse is : If x2 is odd, then x is also odd.
Contrapositive Statement: Contrapositive of “if p then q” is if not q then not p. e.g. contrapositive of
“If a number is divisible by 4, it is divisible by 2”, is
“If a number is not divisible by 2, then it is not divisible by 4”.
Truth Value of “If p then q” : Truth value of the statement “if p then q” is false, when p is true and q is false, otherwise it is true, ue., It is true, when
(i) p is true, q is true.
(it) p is false, q is true.
(iii) p is false, q is false.
However, three methods are adopted to test the truth value of this statement.
(1) Assuming that p is true, prove that q must be true. (Direct method).
(2) Assuming that q is false, prove thatp must be false.
(3) Assuming thatp is true and q is false, obtain a contradiction. (Contradiction method).
Statements with “If and Only If’: “If and only if’, represented by the symbol ⇔, has the following equivalent forms :
(i) p if and only if q.
(it) q if and only p.
(iii) p is necessary and sufficient condition for q and vice-verse.
(iv) p ⇔ q.
Truth Value of “If and only If” : Truth value of statements with “if and only iP is true when p and q are both true or false, otherwise it is false, ue.., statement if and only if is true, when
(i) p is true, q is true.
(ii) p is false, q is false.
Statement with if and one if is false, when
(i) p is true, q is false
(ii) p is false, q is true.
To, Prove A Statement By Contradiction
Example : To prove that √3 is irrational.
Let √3 be rational. So, √3 = p/q, where p and q are co-prime.
p2 = 3q2 ⇒ 3 divides p or 3 is a factor of p.
Let p = 3k. ∴p2 = 9k2.
∴ p2 = 3q2becomes 9k2 = 3q2 or 3k2 = q2.
This means 3 is a factor or q.
i. e., 3 is a factor of p and q both. This is a contradiction.
∴ √3 is irrational.
Counter Example : To prove that a statement is false, we find an example (called counter example) by which the given statement is not valid. Then, we say that the statement is false.