Page 1 NODE6 (E)\DATA\2014\KOTA\JEE-ADVANCED\SMP\MATHS\UNIT#02\ENG\PART-2\03-MATHEMATICAL REASONING\1.THEORY JEE- Mathe ma ti cs 1 6 E MATHEMATICAL REASONING 1 . STATEMENT : A sentence which is either true or false but cannot be both are called a statement. A sentence which is an exclamatory or a wish or an imperative or an interrogative can not be a statement. If a statement is true then its truth value is T and if it is false then its truth value is F For ex. (i) "New Delhi is the capital of India", a true statement (ii) "3 + 2 = 6", a false statement (iii) "Where are you going ?" not a statement beasuse it connot be defined as true or false Note : A statement cannot be both true and false at a time 2 . SIMPLE STATEMENT : Any statement whose truth value does not depend on other statement are called simple statement For ex. (i) " 2 is an irrational number" (ii) "The set of real number is an infinite set" 3 . COMPOUND STATEMENT : A statement which is a combination of two or more simple statements are called compound statement Here the simple statements which form a compound statement are known as its sub statements For ex. (i) "If x is divisible by 2 then x is even number" (ii) " ?ABC is equilatral if and only if its three sides are equal" 4 . LOGICAL CONNECTIVES : The words or phrases which combined simple statements to form a compound statement are called logical connectives. In the following table some possible connectives, their symbols and the nature of the compound statement formed by them S.N. Connectives s y m b o l u s e operation 1. and ? p ? q conjunction 2. or ? p ? q disjunction 3. not ? or ' ? p or p' negation 4. If .... then ..... ??or ? p ??q or p ??q Implication or conditional 5. If and only if (iff) ?? or ? p ??q or p ? ?q Equivalence or Bi-conditional Explanation : (i) p ??q ??statement p and q (p ?? q is true only when p and q both are true otherwise it is false) (ii) p ??q ??statement p or q (p ??q is true if at least one from p and q is true i.e. p ??q is false only when p and q both are false) (iii) ~ p ??not statement p (~ p is true when p is false and ~ p is false when p is true) (iv) p ? ? q ? ?statement p then statement q (p ?? q is false only when p is true and q is false otherwise it is true for all other cases) (v) p ?? q ??statement p if and only if statement q (p ?? q is true only when p and q both are true or false otherwise it is false) JEEMAIN.GURU Page 2 NODE6 (E)\DATA\2014\KOTA\JEE-ADVANCED\SMP\MATHS\UNIT#02\ENG\PART-2\03-MATHEMATICAL REASONING\1.THEORY JEE- Mathe ma ti cs 1 6 E MATHEMATICAL REASONING 1 . STATEMENT : A sentence which is either true or false but cannot be both are called a statement. A sentence which is an exclamatory or a wish or an imperative or an interrogative can not be a statement. If a statement is true then its truth value is T and if it is false then its truth value is F For ex. (i) "New Delhi is the capital of India", a true statement (ii) "3 + 2 = 6", a false statement (iii) "Where are you going ?" not a statement beasuse it connot be defined as true or false Note : A statement cannot be both true and false at a time 2 . SIMPLE STATEMENT : Any statement whose truth value does not depend on other statement are called simple statement For ex. (i) " 2 is an irrational number" (ii) "The set of real number is an infinite set" 3 . COMPOUND STATEMENT : A statement which is a combination of two or more simple statements are called compound statement Here the simple statements which form a compound statement are known as its sub statements For ex. (i) "If x is divisible by 2 then x is even number" (ii) " ?ABC is equilatral if and only if its three sides are equal" 4 . LOGICAL CONNECTIVES : The words or phrases which combined simple statements to form a compound statement are called logical connectives. In the following table some possible connectives, their symbols and the nature of the compound statement formed by them S.N. Connectives s y m b o l u s e operation 1. and ? p ? q conjunction 2. or ? p ? q disjunction 3. not ? or ' ? p or p' negation 4. If .... then ..... ??or ? p ??q or p ??q Implication or conditional 5. If and only if (iff) ?? or ? p ??q or p ? ?q Equivalence or Bi-conditional Explanation : (i) p ??q ??statement p and q (p ?? q is true only when p and q both are true otherwise it is false) (ii) p ??q ??statement p or q (p ??q is true if at least one from p and q is true i.e. p ??q is false only when p and q both are false) (iii) ~ p ??not statement p (~ p is true when p is false and ~ p is false when p is true) (iv) p ? ? q ? ?statement p then statement q (p ?? q is false only when p is true and q is false otherwise it is true for all other cases) (v) p ?? q ??statement p if and only if statement q (p ?? q is true only when p and q both are true or false otherwise it is false) JEEMAIN.GURU NODE6 (E)\DATA\2014\KOTA\JEE-ADVANCED\SMP\MATHS\UNIT#02\ENG\PART-2\03-MATHEMATICAL REASONING\1.THEORY JEE- Mathe ma ti cs E 1 7 5 . TRUTH TABLE : A table which shows the relationship between the truth value of compound statement S(p, q, r ....) and the truth values of its sub statements p, q, r, ... is said to be truth table of compound statement S If p and q are two simple statements then truth table for basic logical connectives are given below Conjunction Disjunction Negation ? p q p q T T T T F F F T F F F F ? p q p q T T T T F T F T T F F F p (~ p) T F F T Conditional Biconditional ? p q p q T T T T F F F T T F F T ? ? ? ? ? ? p q p q q p (p q) (q p) or p q T T T T T T F F T F F T T F F F F T T T Note : If the compound statement contain n sub statements then its truth table will contain 2 n rows. Illustration 1 : Which of the following is correct for the statements p and q ? (1) p ??q is true when at least one from p and q is true (2) p ??q is true when p is true and q is false (3) p ? q is true only when both p and q are true (4) ~ (p ??q) is true only when both p and q are false Solution : We know that p ??q is true only when both p and q are true so option (1) is not correct we know that p ??q is false only when p is true and q is false so option (2) is not correct we know that p ? ?q is true only when either p and q both are true or both are flase so option (3) is not correct we know that ~(p ??q) is true only when (p ??q) is false i.e. p and q both are false So option (4) is correct 6 . LOGICAL EQUIVALENCE : Two compound statements S 1 (p, q, r...) and S 2 (p, q, r ....) are said to be logically equivalent or simply equivalent if they have same truth values for all logically possibilities Two statements S 1 and S 2 are equivalent if they have identical truth table i.e. the entries in the last column of their truth table are same. If statements S 1 and S 2 are equivalent then we write S 1 ? ? S 2 For ex. The truth table for (p ?? q) and (~p ??q) given as below ? ? p q (~ p) p q ~ p q T T F T T T F F F F F T T T T F F T T T We observe that last two columns of the above truth table are identical hence compound statements (p ?? q) and (~p ??q) are equivalent i.e. ? ? ? p q ~ p q JEEMAIN.GURU Page 3 NODE6 (E)\DATA\2014\KOTA\JEE-ADVANCED\SMP\MATHS\UNIT#02\ENG\PART-2\03-MATHEMATICAL REASONING\1.THEORY JEE- Mathe ma ti cs 1 6 E MATHEMATICAL REASONING 1 . STATEMENT : A sentence which is either true or false but cannot be both are called a statement. A sentence which is an exclamatory or a wish or an imperative or an interrogative can not be a statement. If a statement is true then its truth value is T and if it is false then its truth value is F For ex. (i) "New Delhi is the capital of India", a true statement (ii) "3 + 2 = 6", a false statement (iii) "Where are you going ?" not a statement beasuse it connot be defined as true or false Note : A statement cannot be both true and false at a time 2 . SIMPLE STATEMENT : Any statement whose truth value does not depend on other statement are called simple statement For ex. (i) " 2 is an irrational number" (ii) "The set of real number is an infinite set" 3 . COMPOUND STATEMENT : A statement which is a combination of two or more simple statements are called compound statement Here the simple statements which form a compound statement are known as its sub statements For ex. (i) "If x is divisible by 2 then x is even number" (ii) " ?ABC is equilatral if and only if its three sides are equal" 4 . LOGICAL CONNECTIVES : The words or phrases which combined simple statements to form a compound statement are called logical connectives. In the following table some possible connectives, their symbols and the nature of the compound statement formed by them S.N. Connectives s y m b o l u s e operation 1. and ? p ? q conjunction 2. or ? p ? q disjunction 3. not ? or ' ? p or p' negation 4. If .... then ..... ??or ? p ??q or p ??q Implication or conditional 5. If and only if (iff) ?? or ? p ??q or p ? ?q Equivalence or Bi-conditional Explanation : (i) p ??q ??statement p and q (p ?? q is true only when p and q both are true otherwise it is false) (ii) p ??q ??statement p or q (p ??q is true if at least one from p and q is true i.e. p ??q is false only when p and q both are false) (iii) ~ p ??not statement p (~ p is true when p is false and ~ p is false when p is true) (iv) p ? ? q ? ?statement p then statement q (p ?? q is false only when p is true and q is false otherwise it is true for all other cases) (v) p ?? q ??statement p if and only if statement q (p ?? q is true only when p and q both are true or false otherwise it is false) JEEMAIN.GURU NODE6 (E)\DATA\2014\KOTA\JEE-ADVANCED\SMP\MATHS\UNIT#02\ENG\PART-2\03-MATHEMATICAL REASONING\1.THEORY JEE- Mathe ma ti cs E 1 7 5 . TRUTH TABLE : A table which shows the relationship between the truth value of compound statement S(p, q, r ....) and the truth values of its sub statements p, q, r, ... is said to be truth table of compound statement S If p and q are two simple statements then truth table for basic logical connectives are given below Conjunction Disjunction Negation ? p q p q T T T T F F F T F F F F ? p q p q T T T T F T F T T F F F p (~ p) T F F T Conditional Biconditional ? p q p q T T T T F F F T T F F T ? ? ? ? ? ? p q p q q p (p q) (q p) or p q T T T T T T F F T F F T T F F F F T T T Note : If the compound statement contain n sub statements then its truth table will contain 2 n rows. Illustration 1 : Which of the following is correct for the statements p and q ? (1) p ??q is true when at least one from p and q is true (2) p ??q is true when p is true and q is false (3) p ? q is true only when both p and q are true (4) ~ (p ??q) is true only when both p and q are false Solution : We know that p ??q is true only when both p and q are true so option (1) is not correct we know that p ??q is false only when p is true and q is false so option (2) is not correct we know that p ? ?q is true only when either p and q both are true or both are flase so option (3) is not correct we know that ~(p ??q) is true only when (p ??q) is false i.e. p and q both are false So option (4) is correct 6 . LOGICAL EQUIVALENCE : Two compound statements S 1 (p, q, r...) and S 2 (p, q, r ....) are said to be logically equivalent or simply equivalent if they have same truth values for all logically possibilities Two statements S 1 and S 2 are equivalent if they have identical truth table i.e. the entries in the last column of their truth table are same. If statements S 1 and S 2 are equivalent then we write S 1 ? ? S 2 For ex. The truth table for (p ?? q) and (~p ??q) given as below ? ? p q (~ p) p q ~ p q T T F T T T F F F F F T T T T F F T T T We observe that last two columns of the above truth table are identical hence compound statements (p ?? q) and (~p ??q) are equivalent i.e. ? ? ? p q ~ p q JEEMAIN.GURU NODE6 (E)\DATA\2014\KOTA\JEE-ADVANCED\SMP\MATHS\UNIT#02\ENG\PART-2\03-MATHEMATICAL REASONING\1.THEORY JEE- Mathe ma ti cs 1 8 E Illustration 2 : Equivalent statement of the statement "if 8 > 10 then 2 2 = 5" will be :- (1) if 2 2 = 5 then 8 > 10 (2) 8 < 10 and 2 2 ? 5 (3) 8 < 10 or 2 2 = 5 (4) none of these Solution : We know that p ? q ? ~p ??q ? ? ? equivalent statment will ? ? 8 10 or 2 2 = 5 or 8 ? 10 or 2 2 = 5 So (4) will be the correct answer. Do yourself - 1 : ( i ) Which of the following is logically equivalent to (p ? ? q) ? (1) p ??~q (2) ~p ? ~ q (3) ~(p ??~q) (4) ~(~p ??~q) 7 . TAUTOLOGY AND CONTRADICTION : ( i ) Tautology : A statement is said to be a tautology if it is true for all logical possibilities i.e. its truth value always T. it is denoted by t. For ex. the statement p ?? ~ (p ??q) is a tautology ? ? ? ? p q p q ~ (p q) p ~ (p q) T T T F T T F F T T F T F T T F F F T T Clearly, The truth value of p ??~ (p ??q) is T for all values of p and q. so p ??~ (p ??q) is a tautology (i i) Contradiction : A statement is a contradiction if it is false for all logical possibilities. i.e. its truth value always F. It is denoted by c. For ex. The statement (p ?? q) ? ? (~p ?? ~q) is a contradiction ? ? ? ? ? p q ~ p ~ q p q (~ p ~ q) (p q) (~ p ~ q) T T F F T F F T F F T T F F F T T F T F F F F T T F T F Clearly, then truth value of (p ? ? q) ? ? (~p ? ? ~q) is F for all value of p and q. So (p ? ? q) ? ? (~p ? ? ~q) is a contradiction. Note : The negation of a tautology is a contradiction and negation of a contradiction is a tautology Do yourself - 2 : By truth table prove that : ( i ) p ? q ? ~p ? ~q (ii) ? ? ? ? p (~ p q) p q (iii) ? ? p (~ p q) is a tautology. 8 . ALGEBR A OF STATEMENTS : If p, q, r are any three statements then the some low of algebra of statements are as follow ( i ) Idempotent Laws : (a) p ??p ??p (b) p ??p ??p i.e. p ??p ??p ??p ??p ? ? p (p p) (p p) T T T F F F JEEMAIN.GURU Page 4 NODE6 (E)\DATA\2014\KOTA\JEE-ADVANCED\SMP\MATHS\UNIT#02\ENG\PART-2\03-MATHEMATICAL REASONING\1.THEORY JEE- Mathe ma ti cs 1 6 E MATHEMATICAL REASONING 1 . STATEMENT : A sentence which is either true or false but cannot be both are called a statement. A sentence which is an exclamatory or a wish or an imperative or an interrogative can not be a statement. If a statement is true then its truth value is T and if it is false then its truth value is F For ex. (i) "New Delhi is the capital of India", a true statement (ii) "3 + 2 = 6", a false statement (iii) "Where are you going ?" not a statement beasuse it connot be defined as true or false Note : A statement cannot be both true and false at a time 2 . SIMPLE STATEMENT : Any statement whose truth value does not depend on other statement are called simple statement For ex. (i) " 2 is an irrational number" (ii) "The set of real number is an infinite set" 3 . COMPOUND STATEMENT : A statement which is a combination of two or more simple statements are called compound statement Here the simple statements which form a compound statement are known as its sub statements For ex. (i) "If x is divisible by 2 then x is even number" (ii) " ?ABC is equilatral if and only if its three sides are equal" 4 . LOGICAL CONNECTIVES : The words or phrases which combined simple statements to form a compound statement are called logical connectives. In the following table some possible connectives, their symbols and the nature of the compound statement formed by them S.N. Connectives s y m b o l u s e operation 1. and ? p ? q conjunction 2. or ? p ? q disjunction 3. not ? or ' ? p or p' negation 4. If .... then ..... ??or ? p ??q or p ??q Implication or conditional 5. If and only if (iff) ?? or ? p ??q or p ? ?q Equivalence or Bi-conditional Explanation : (i) p ??q ??statement p and q (p ?? q is true only when p and q both are true otherwise it is false) (ii) p ??q ??statement p or q (p ??q is true if at least one from p and q is true i.e. p ??q is false only when p and q both are false) (iii) ~ p ??not statement p (~ p is true when p is false and ~ p is false when p is true) (iv) p ? ? q ? ?statement p then statement q (p ?? q is false only when p is true and q is false otherwise it is true for all other cases) (v) p ?? q ??statement p if and only if statement q (p ?? q is true only when p and q both are true or false otherwise it is false) JEEMAIN.GURU NODE6 (E)\DATA\2014\KOTA\JEE-ADVANCED\SMP\MATHS\UNIT#02\ENG\PART-2\03-MATHEMATICAL REASONING\1.THEORY JEE- Mathe ma ti cs E 1 7 5 . TRUTH TABLE : A table which shows the relationship between the truth value of compound statement S(p, q, r ....) and the truth values of its sub statements p, q, r, ... is said to be truth table of compound statement S If p and q are two simple statements then truth table for basic logical connectives are given below Conjunction Disjunction Negation ? p q p q T T T T F F F T F F F F ? p q p q T T T T F T F T T F F F p (~ p) T F F T Conditional Biconditional ? p q p q T T T T F F F T T F F T ? ? ? ? ? ? p q p q q p (p q) (q p) or p q T T T T T T F F T F F T T F F F F T T T Note : If the compound statement contain n sub statements then its truth table will contain 2 n rows. Illustration 1 : Which of the following is correct for the statements p and q ? (1) p ??q is true when at least one from p and q is true (2) p ??q is true when p is true and q is false (3) p ? q is true only when both p and q are true (4) ~ (p ??q) is true only when both p and q are false Solution : We know that p ??q is true only when both p and q are true so option (1) is not correct we know that p ??q is false only when p is true and q is false so option (2) is not correct we know that p ? ?q is true only when either p and q both are true or both are flase so option (3) is not correct we know that ~(p ??q) is true only when (p ??q) is false i.e. p and q both are false So option (4) is correct 6 . LOGICAL EQUIVALENCE : Two compound statements S 1 (p, q, r...) and S 2 (p, q, r ....) are said to be logically equivalent or simply equivalent if they have same truth values for all logically possibilities Two statements S 1 and S 2 are equivalent if they have identical truth table i.e. the entries in the last column of their truth table are same. If statements S 1 and S 2 are equivalent then we write S 1 ? ? S 2 For ex. The truth table for (p ?? q) and (~p ??q) given as below ? ? p q (~ p) p q ~ p q T T F T T T F F F F F T T T T F F T T T We observe that last two columns of the above truth table are identical hence compound statements (p ?? q) and (~p ??q) are equivalent i.e. ? ? ? p q ~ p q JEEMAIN.GURU NODE6 (E)\DATA\2014\KOTA\JEE-ADVANCED\SMP\MATHS\UNIT#02\ENG\PART-2\03-MATHEMATICAL REASONING\1.THEORY JEE- Mathe ma ti cs 1 8 E Illustration 2 : Equivalent statement of the statement "if 8 > 10 then 2 2 = 5" will be :- (1) if 2 2 = 5 then 8 > 10 (2) 8 < 10 and 2 2 ? 5 (3) 8 < 10 or 2 2 = 5 (4) none of these Solution : We know that p ? q ? ~p ??q ? ? ? equivalent statment will ? ? 8 10 or 2 2 = 5 or 8 ? 10 or 2 2 = 5 So (4) will be the correct answer. Do yourself - 1 : ( i ) Which of the following is logically equivalent to (p ? ? q) ? (1) p ??~q (2) ~p ? ~ q (3) ~(p ??~q) (4) ~(~p ??~q) 7 . TAUTOLOGY AND CONTRADICTION : ( i ) Tautology : A statement is said to be a tautology if it is true for all logical possibilities i.e. its truth value always T. it is denoted by t. For ex. the statement p ?? ~ (p ??q) is a tautology ? ? ? ? p q p q ~ (p q) p ~ (p q) T T T F T T F F T T F T F T T F F F T T Clearly, The truth value of p ??~ (p ??q) is T for all values of p and q. so p ??~ (p ??q) is a tautology (i i) Contradiction : A statement is a contradiction if it is false for all logical possibilities. i.e. its truth value always F. It is denoted by c. For ex. The statement (p ?? q) ? ? (~p ?? ~q) is a contradiction ? ? ? ? ? p q ~ p ~ q p q (~ p ~ q) (p q) (~ p ~ q) T T F F T F F T F F T T F F F T T F T F F F F T T F T F Clearly, then truth value of (p ? ? q) ? ? (~p ? ? ~q) is F for all value of p and q. So (p ? ? q) ? ? (~p ? ? ~q) is a contradiction. Note : The negation of a tautology is a contradiction and negation of a contradiction is a tautology Do yourself - 2 : By truth table prove that : ( i ) p ? q ? ~p ? ~q (ii) ? ? ? ? p (~ p q) p q (iii) ? ? p (~ p q) is a tautology. 8 . ALGEBR A OF STATEMENTS : If p, q, r are any three statements then the some low of algebra of statements are as follow ( i ) Idempotent Laws : (a) p ??p ??p (b) p ??p ??p i.e. p ??p ??p ??p ??p ? ? p (p p) (p p) T T T F F F JEEMAIN.GURU NODE6 (E)\DATA\2014\KOTA\JEE-ADVANCED\SMP\MATHS\UNIT#02\ENG\PART-2\03-MATHEMATICAL REASONING\1.THEORY JEE- Mathe ma ti cs E 1 9 (i i) Comutative laws : (a) p ??q ??q ??p (b) p ??q ??q ??p ? ? ? ? p q (p q) (q p) (p q) (q p) T T T T T T T F F F T T F T F F T T F F F F F F (iii) Associative laws : (a) (p ??q) ??r ??p ??(q ??r) (b) (p ??q) ??r ??p ??(q ??r) ? ? ? ? ? ? p q r (p q) (q r) (p q) r p (q r) T T T T T T T T T F T F F F T F T F F F F T F F F F F F F T T F T F F F T F F F F F F F T F F F F F F F F F F F Similarly we can proved result (b) ( i v) Distributive laws : (a) p ??(q ??r) ??(p ??q) ??(p ??r) (c) p ? (q ??r) ? (p ??q) ? (p ??r) (b) p ??(q ??r) ??(p ??q) ??(p ??r) (d) p ??(q ??r) ? (p ??q) ??(p ??r) ? ? ? ? ? ? ? ? p q r (q r) (p q) (p r) p (q r) (p q) (p r) T T T T T T T T T T F T T F T T T F T T F T T T T F F F F F F F F T T T F F F F F T F T F F F F F F T T F F F F F F F F F F F F Similarly we can prove result (b), (c), (d) ( v ) De Morgan Laws : (a) ~ (p ? ? q) ? ? ~p ? ? ~q (b) ~(p ??q) ??~p ??~q ? ? ? p q ~ p ~ q (p q) ~ (p q) (~ p ~ q) T T F F T F F T F F T F T T F T T F F T T F F T T F T T Similarly we can proved resulty (b) ( v i ) Involution laws (or Double negation laws) : ~(~p) ? ? p p ~ p ~ (~ p) T F T F T F JEEMAIN.GURU Page 5 NODE6 (E)\DATA\2014\KOTA\JEE-ADVANCED\SMP\MATHS\UNIT#02\ENG\PART-2\03-MATHEMATICAL REASONING\1.THEORY JEE- Mathe ma ti cs 1 6 E MATHEMATICAL REASONING 1 . STATEMENT : A sentence which is either true or false but cannot be both are called a statement. A sentence which is an exclamatory or a wish or an imperative or an interrogative can not be a statement. If a statement is true then its truth value is T and if it is false then its truth value is F For ex. (i) "New Delhi is the capital of India", a true statement (ii) "3 + 2 = 6", a false statement (iii) "Where are you going ?" not a statement beasuse it connot be defined as true or false Note : A statement cannot be both true and false at a time 2 . SIMPLE STATEMENT : Any statement whose truth value does not depend on other statement are called simple statement For ex. (i) " 2 is an irrational number" (ii) "The set of real number is an infinite set" 3 . COMPOUND STATEMENT : A statement which is a combination of two or more simple statements are called compound statement Here the simple statements which form a compound statement are known as its sub statements For ex. (i) "If x is divisible by 2 then x is even number" (ii) " ?ABC is equilatral if and only if its three sides are equal" 4 . LOGICAL CONNECTIVES : The words or phrases which combined simple statements to form a compound statement are called logical connectives. In the following table some possible connectives, their symbols and the nature of the compound statement formed by them S.N. Connectives s y m b o l u s e operation 1. and ? p ? q conjunction 2. or ? p ? q disjunction 3. not ? or ' ? p or p' negation 4. If .... then ..... ??or ? p ??q or p ??q Implication or conditional 5. If and only if (iff) ?? or ? p ??q or p ? ?q Equivalence or Bi-conditional Explanation : (i) p ??q ??statement p and q (p ?? q is true only when p and q both are true otherwise it is false) (ii) p ??q ??statement p or q (p ??q is true if at least one from p and q is true i.e. p ??q is false only when p and q both are false) (iii) ~ p ??not statement p (~ p is true when p is false and ~ p is false when p is true) (iv) p ? ? q ? ?statement p then statement q (p ?? q is false only when p is true and q is false otherwise it is true for all other cases) (v) p ?? q ??statement p if and only if statement q (p ?? q is true only when p and q both are true or false otherwise it is false) JEEMAIN.GURU NODE6 (E)\DATA\2014\KOTA\JEE-ADVANCED\SMP\MATHS\UNIT#02\ENG\PART-2\03-MATHEMATICAL REASONING\1.THEORY JEE- Mathe ma ti cs E 1 7 5 . TRUTH TABLE : A table which shows the relationship between the truth value of compound statement S(p, q, r ....) and the truth values of its sub statements p, q, r, ... is said to be truth table of compound statement S If p and q are two simple statements then truth table for basic logical connectives are given below Conjunction Disjunction Negation ? p q p q T T T T F F F T F F F F ? p q p q T T T T F T F T T F F F p (~ p) T F F T Conditional Biconditional ? p q p q T T T T F F F T T F F T ? ? ? ? ? ? p q p q q p (p q) (q p) or p q T T T T T T F F T F F T T F F F F T T T Note : If the compound statement contain n sub statements then its truth table will contain 2 n rows. Illustration 1 : Which of the following is correct for the statements p and q ? (1) p ??q is true when at least one from p and q is true (2) p ??q is true when p is true and q is false (3) p ? q is true only when both p and q are true (4) ~ (p ??q) is true only when both p and q are false Solution : We know that p ??q is true only when both p and q are true so option (1) is not correct we know that p ??q is false only when p is true and q is false so option (2) is not correct we know that p ? ?q is true only when either p and q both are true or both are flase so option (3) is not correct we know that ~(p ??q) is true only when (p ??q) is false i.e. p and q both are false So option (4) is correct 6 . LOGICAL EQUIVALENCE : Two compound statements S 1 (p, q, r...) and S 2 (p, q, r ....) are said to be logically equivalent or simply equivalent if they have same truth values for all logically possibilities Two statements S 1 and S 2 are equivalent if they have identical truth table i.e. the entries in the last column of their truth table are same. If statements S 1 and S 2 are equivalent then we write S 1 ? ? S 2 For ex. The truth table for (p ?? q) and (~p ??q) given as below ? ? p q (~ p) p q ~ p q T T F T T T F F F F F T T T T F F T T T We observe that last two columns of the above truth table are identical hence compound statements (p ?? q) and (~p ??q) are equivalent i.e. ? ? ? p q ~ p q JEEMAIN.GURU NODE6 (E)\DATA\2014\KOTA\JEE-ADVANCED\SMP\MATHS\UNIT#02\ENG\PART-2\03-MATHEMATICAL REASONING\1.THEORY JEE- Mathe ma ti cs 1 8 E Illustration 2 : Equivalent statement of the statement "if 8 > 10 then 2 2 = 5" will be :- (1) if 2 2 = 5 then 8 > 10 (2) 8 < 10 and 2 2 ? 5 (3) 8 < 10 or 2 2 = 5 (4) none of these Solution : We know that p ? q ? ~p ??q ? ? ? equivalent statment will ? ? 8 10 or 2 2 = 5 or 8 ? 10 or 2 2 = 5 So (4) will be the correct answer. Do yourself - 1 : ( i ) Which of the following is logically equivalent to (p ? ? q) ? (1) p ??~q (2) ~p ? ~ q (3) ~(p ??~q) (4) ~(~p ??~q) 7 . TAUTOLOGY AND CONTRADICTION : ( i ) Tautology : A statement is said to be a tautology if it is true for all logical possibilities i.e. its truth value always T. it is denoted by t. For ex. the statement p ?? ~ (p ??q) is a tautology ? ? ? ? p q p q ~ (p q) p ~ (p q) T T T F T T F F T T F T F T T F F F T T Clearly, The truth value of p ??~ (p ??q) is T for all values of p and q. so p ??~ (p ??q) is a tautology (i i) Contradiction : A statement is a contradiction if it is false for all logical possibilities. i.e. its truth value always F. It is denoted by c. For ex. The statement (p ?? q) ? ? (~p ?? ~q) is a contradiction ? ? ? ? ? p q ~ p ~ q p q (~ p ~ q) (p q) (~ p ~ q) T T F F T F F T F F T T F F F T T F T F F F F T T F T F Clearly, then truth value of (p ? ? q) ? ? (~p ? ? ~q) is F for all value of p and q. So (p ? ? q) ? ? (~p ? ? ~q) is a contradiction. Note : The negation of a tautology is a contradiction and negation of a contradiction is a tautology Do yourself - 2 : By truth table prove that : ( i ) p ? q ? ~p ? ~q (ii) ? ? ? ? p (~ p q) p q (iii) ? ? p (~ p q) is a tautology. 8 . ALGEBR A OF STATEMENTS : If p, q, r are any three statements then the some low of algebra of statements are as follow ( i ) Idempotent Laws : (a) p ??p ??p (b) p ??p ??p i.e. p ??p ??p ??p ??p ? ? p (p p) (p p) T T T F F F JEEMAIN.GURU NODE6 (E)\DATA\2014\KOTA\JEE-ADVANCED\SMP\MATHS\UNIT#02\ENG\PART-2\03-MATHEMATICAL REASONING\1.THEORY JEE- Mathe ma ti cs E 1 9 (i i) Comutative laws : (a) p ??q ??q ??p (b) p ??q ??q ??p ? ? ? ? p q (p q) (q p) (p q) (q p) T T T T T T T F F F T T F T F F T T F F F F F F (iii) Associative laws : (a) (p ??q) ??r ??p ??(q ??r) (b) (p ??q) ??r ??p ??(q ??r) ? ? ? ? ? ? p q r (p q) (q r) (p q) r p (q r) T T T T T T T T T F T F F F T F T F F F F T F F F F F F F T T F T F F F T F F F F F F F T F F F F F F F F F F F Similarly we can proved result (b) ( i v) Distributive laws : (a) p ??(q ??r) ??(p ??q) ??(p ??r) (c) p ? (q ??r) ? (p ??q) ? (p ??r) (b) p ??(q ??r) ??(p ??q) ??(p ??r) (d) p ??(q ??r) ? (p ??q) ??(p ??r) ? ? ? ? ? ? ? ? p q r (q r) (p q) (p r) p (q r) (p q) (p r) T T T T T T T T T T F T T F T T T F T T F T T T T F F F F F F F F T T T F F F F F T F T F F F F F F T T F F F F F F F F F F F F Similarly we can prove result (b), (c), (d) ( v ) De Morgan Laws : (a) ~ (p ? ? q) ? ? ~p ? ? ~q (b) ~(p ??q) ??~p ??~q ? ? ? p q ~ p ~ q (p q) ~ (p q) (~ p ~ q) T T F F T F F T F F T F T T F T T F F T T F F T T F T T Similarly we can proved resulty (b) ( v i ) Involution laws (or Double negation laws) : ~(~p) ? ? p p ~ p ~ (~ p) T F T F T F JEEMAIN.GURU NODE6 (E)\DATA\2014\KOTA\JEE-ADVANCED\SMP\MATHS\UNIT#02\ENG\PART-2\03-MATHEMATICAL REASONING\1.THEORY JEE- Mathe ma ti cs 2 0 E (vii) Identity Laws : If p is a statement and t and c are tautology and contradiction respectively then (a) p ??t ??p (b) p ??t ??t (c) p ??c ??c (d) p ??c ??p ? ? ? ? p t c (p t) (p t) (p c) (p c) T T F T T F T F T F F T F F (viii) Complement Laws : (a) p ??(~p) ??c (b) p ??(~p) ??t (c) (~t) ??c (d) (~c) ??t ? ? p ~ p (p ~ p) (p ~ p) T F F T F T F T ( ix ) Contrapositive laws : p ? ? q ? ? ~q ? ? ~p ? ? p q ~ p ~ q p q ~ q ~ p T T F F T T T F F T F F F T T F T T F F T T T T Illustration 3 : ~(p ??q) ??(~p ??q) is equivalent to- (1) p (2) ~p (3) q (4) ~q Solution : ? ~(p ??q) ??(~p ??q) ??(~p ??~q) ??(~p ??q) (By Demorgan Law) ??~p ??(~q ??q) (By distributive laws) ??~p ??t (By complement laws) ??~p (By Identity Laws) Ans. (2) Do yourself - 3 : ( i ) Statement (p ??~q) ??(~p ??q) is (1) a tautology (2) a contradiction (3) neither a tautology not a contradiction (4) None of these 9 . NEGATION OF COMPOUND STATEMENTS : If p and q are two statements then (i) Negation of conjunction : ~(p ? ? q) ? ? ~p ? ? ~q ? ? ? p q ~ p ~ q (p q) ~ (p q) (~ p ~ q) T T F F T F F T F F T F T T F T T F F T T F F T T F T T (i i) Negation of disjunction : ~(p ? ? q) ? ? ~p ? ? ~q ? ? ? p q ~ p ~ q (p q) (~ p q) (~ p ~ q) T T F F T F F T F F T T F F F T T F T F F F F T T F T T JEEMAIN.GURURead More

Offer running on EduRev: __Apply code STAYHOME200__ to get INR 200 off on our premium plan EduRev Infinity!