Mathematical Reasoning JEE Notes | EduRev

JEE : Mathematical Reasoning JEE Notes | EduRev

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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
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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
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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
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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
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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
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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
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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)
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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
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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
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(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
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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)
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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
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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
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(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
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(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
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