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.