Notes On Validating Statements - CBSE Class 11 Maths
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, x2 is a negative number. (False) If x is odd, then x2 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 — 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 x2 is odd. The component statements: p: x is odd. q: x2 is odd. Consider p: x is odd, is true. x = 2n + 1, for some integer n. x2 = (2n + 1)2     = 4n2 + 4n + 1     = 2(2n2 + 2n) + 1     = 2l + 1, where l = 2n2 + 2n Hence, x2 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) ≠ 9 + x2 ⇒ (3 + 0)(3 - 0) ≠ 9 + 0 ⇒ 9 ≠ 9 On simplification, 9 ≠ 9, which is a contradiction. Hence, the given statement is false.

#### Summary

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, x2 is a negative number. (False) If x is odd, then x2 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 — 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 x2 is odd. The component statements: p: x is odd. q: x2 is odd. Consider p: x is odd, is true. x = 2n + 1, for some integer n. x2 = (2n + 1)2     = 4n2 + 4n + 1     = 2(2n2 + 2n) + 1     = 2l + 1, where l = 2n2 + 2n Hence, x2 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) ≠ 9 + x2 ⇒ (3 + 0)(3 - 0) ≠ 9 + 0 ⇒ 9 ≠ 9 On simplification, 9 ≠ 9, which is a contradiction. Hence, the given statement is false.

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