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Every mathematical statement is either true of false.

Ex:

2 is a positive number and a prime number. (True)

5 is a factor of 25 or 30. (True)

For every integer x, x^{2} is a negative number. (False)

If x is odd, then x^{2} is odd. (True)

x - 2 is negative if and only if x is less than 2. (True)

The truth value of a compound statement depends on the truth values of the component statements involved in it.

To validate the component statements, certain rules are followed.

**Rule I:** The first rule is used to validate compound statements with the connective word "and," or conjunctions. To validate whether a conjunction is true, we have to show that all of its component statements are true.

For mathematical statement s p and q, to show that the statement "p and q" is true, we need to show both p and q are true.

Ex: Validity of the statement â€” 2 is a positive number and a prime number.

The component statements:

p: 2 is a positive number.

q: 2 is a prime number.

Component statements p and q are true.

Hence, the given compound statement is true.

**Rule 2:** The second rule is used to validate compound statements with the connective word "or".

To validate whether a disjunction is true, we have to show that one of its component statements is true.

If p and q are mathematical statements, then in order to show that the statement "p or q" is true, we need to show that either statement p is true or statement q is true.

Ex: Check the validity of the statement, "Five is a factor of 21 or 30."

The component statements:

p: 5 is a factor of 21.

q: 5 is a factor of 30.

The statement p is false, while the statement q is true.

Hence, it is validated that the given compound statement is true.

**Rule III:** Rule three is used to validate statements with "if-then," or implications.

To prove the statement "if p, then q", we need to show any one of the following cases is true.

Case 1: By assuming that p is true, show that q must be true. (Direct method)

Case 2: By assuming that q is false, show that p must be false. (Contrapositive method)

Example: If x is odd, then x^{2} is odd.

The component statements:

p: x is odd.

q: x^{2} is odd.

Consider p: x is odd, is true.

x = 2n + 1, for some integer n.

x^{2} = (2n + 1)^{2}

= 4n^{2} + 4n + 1

= 2(2n^{2} + 2n) + 1

= 2l + 1, where l = 2n^{2} + 2n

Hence, x^{2} is odd.

Hence, the given statement is proved.

**Rule IV:** Rule four is used to prove bi-implications.

To prove the statement "p if and only if q," we need to show the following.

If p is true, then q is true.

If q is true, then p is true.

Example: x - 2 is negative if and only if x is less than 2.

The component statements are

If x - 2 is negative, then x is less than 2.

If x is less than 2, then x - 2 is negative.

Both the implications are true.

Therefore, the bi-implication 'x - 2 is negative if and only if x is less than 2.' is true.

**Contradiction Method**

Some mathematical statements cannot be proved directly.

Such statements can be proved by the contradiction method.

Contradiction Method: To prove that a statement p is true, first, assume that p is not true or negation p is true. Then, we arrive at some result that contradicts the assumption. This concludes that p is true.

Consider the example of a situation where the statement is not valid. Such examples are called counter examples.

Counter example: The example of a situation where a statement is not valid.

Put x = 0,

(3 + x)(3 - x) â‰ 9 + x^{2}

â‡’ (3 + 0)(3 - 0) â‰ 9 + 0

â‡’ 9 â‰ 9

On simplification, 9 â‰ 9, which is a contradiction.

Hence, the given statement is false.

Every mathematical statement is either true of false.

Ex:

2 is a positive number and a prime number. (True)

5 is a factor of 25 or 30. (True)

For every integer x, x^{2} is a negative number. (False)

If x is odd, then x^{2} is odd. (True)

x - 2 is negative if and only if x is less than 2. (True)

The truth value of a compound statement depends on the truth values of the component statements involved in it.

To validate the component statements, certain rules are followed.

**Rule I:** The first rule is used to validate compound statements with the connective word "and," or conjunctions. To validate whether a conjunction is true, we have to show that all of its component statements are true.

For mathematical statement s p and q, to show that the statement "p and q" is true, we need to show both p and q are true.

Ex: Validity of the statement â€” 2 is a positive number and a prime number.

The component statements:

p: 2 is a positive number.

q: 2 is a prime number.

Component statements p and q are true.

Hence, the given compound statement is true.

**Rule 2:** The second rule is used to validate compound statements with the connective word "or".

To validate whether a disjunction is true, we have to show that one of its component statements is true.

If p and q are mathematical statements, then in order to show that the statement "p or q" is true, we need to show that either statement p is true or statement q is true.

Ex: Check the validity of the statement, "Five is a factor of 21 or 30."

The component statements:

p: 5 is a factor of 21.

q: 5 is a factor of 30.

The statement p is false, while the statement q is true.

Hence, it is validated that the given compound statement is true.

**Rule III:** Rule three is used to validate statements with "if-then," or implications.

To prove the statement "if p, then q", we need to show any one of the following cases is true.

Case 1: By assuming that p is true, show that q must be true. (Direct method)

Case 2: By assuming that q is false, show that p must be false. (Contrapositive method)

Example: If x is odd, then x^{2} is odd.

The component statements:

p: x is odd.

q: x^{2} is odd.

Consider p: x is odd, is true.

x = 2n + 1, for some integer n.

x^{2} = (2n + 1)^{2}

= 4n^{2} + 4n + 1

= 2(2n^{2} + 2n) + 1

= 2l + 1, where l = 2n^{2} + 2n

Hence, x^{2} is odd.

Hence, the given statement is proved.

**Rule IV:** Rule four is used to prove bi-implications.

To prove the statement "p if and only if q," we need to show the following.

If p is true, then q is true.

If q is true, then p is true.

Example: x - 2 is negative if and only if x is less than 2.

The component statements are

If x - 2 is negative, then x is less than 2.

If x is less than 2, then x - 2 is negative.

Both the implications are true.

Therefore, the bi-implication 'x - 2 is negative if and only if x is less than 2.' is true.

**Contradiction Method**

Some mathematical statements cannot be proved directly.

Such statements can be proved by the contradiction method.

Contradiction Method: To prove that a statement p is true, first, assume that p is not true or negation p is true. Then, we arrive at some result that contradicts the assumption. This concludes that p is true.

Consider the example of a situation where the statement is not valid. Such examples are called counter examples.

Counter example: The example of a situation where a statement is not valid.

Put x = 0,

(3 + x)(3 - x) â‰ 9 + x^{2}

â‡’ (3 + 0)(3 - 0) â‰ 9 + 0

â‡’ 9 â‰ 9

On simplification, 9 â‰ 9, which is a contradiction.

Hence, the given statement is false.