MATHS 1 – SCIENCE (Mind Faces)
(Quick Guidance CHEAT SHEET)
Note – Questions given here are indicative and typical. However, in exams, only
these kind of questions will be asked
1. Mathematical Logic
1.1 Statement :
A statement is a declarative (assertive) sentence which is either true or false, but not both
simultaneously. Statements are denoted by p, q, r, .....
Truth value of a statement :
Each statement is either true or false. If a statement is true then its truth value is 'T' and
if the statement is false then its truth value is F.
Following sentences are statements.
i) 5 2 = 11
ii) Every triangle has three sides.
iii) Mumbai is the capital of Maharashtra.
Following sentences are not statements.
i) Please, give your Pen.
ii) What is your name?
iii) Open the window.
Note : Interrogative, exclamatory, command, order, request, suggestion are not statements.
Let us learn these statements :
i) For x = 6 it is true but for other than 6 it is not true. Here, we cannot determine the truth
value.
ii) For ‘It is black in colour.’ the truth value varies from person to person. In the above
sentences, the truth value depends upon the situation. Such sentences are called as open
sentences. Open sentence is not a statement.
Logical connectives, simple and compound statements :
The words or phrases which are used to connect two statements are called logical
connectives.
We will study the connectives 'and', 'or', 'if ..... then', 'if and only if ', 'not''.
Simple and Compound Statements : A statement which cannot be split further into two
or more statements is called a simple statement. If a statement is the combination of two
or more simple statements, then it is called a compound statement.
"3 is a prime and 4 is an even number", is a compound statement.
"3 and 5 are twin primes", is a simple statement.
,1.2 Statement Pattern, Logical Equivalence, Tautology, Contradiction, Contingency.
1) Statement Pattern : Letters used to denote statements are called statement letters.
Proper combination of statement letters and connectives is called a statement pattern.
Statement pattern is also called as a proposition. p → q, p q, p q are statement
patterns. p and q are their prime components.
A table which shows the possible truth values of a statement pattern obtained by
considering all possible combinations of truth values of its prime components is called
the truth table of the statement pattern.
2) Logical Equivalence :
Two statement patterns are said to be equivalent if their truth tables are identical. If
statement patterns A and B are equivalent, we write it as A B.
3) Tautology, Contradiction and Contingency :
⚫ Tautology : A statement pattern whose truth value is true for all possible combinations
of truth values of its prime components is called a tautology. We denote tautology
by t.
Ex. Statement pattern p p is a tautology.
⚫ Contradiction : A statement pattern whose truth value is false for all possible
combinations of truth values of its prime components is called a contradiction. We
denote contradiction by c.
Ex. Statement pattern p p is a contradiction.
⚫ Contingency : A statement pattern which is neither a tautology nor a contradiction
is called a contingency.
Ex. p q is a contingency.
Ex. 1) [(p q) r] (p (q r)]
p q r p q (p q) r q r p (q r) [(p q) r] [p (q r)]
T T T T T T T T
T T F T T T T T
T F T T T T T T
T F F T T F T T
F T T T T T T T
F T F T T T T T
F F T F T T T T
F F F F F F F T
All the truth values in the last column are T, hence it is tautology.
Standard - XII : Subject - Mathematics and Statistics (Arts and Science) Part - I : 2
,Ex. 2) [p p → q)] → q
p q q p → q p ( p → q p (p → q) → q
T T F F F T
T F T T T F
F T F T F T
F F T T F T
Truth values in the last column are not identical. Hence it is contingency.
Ex. 3) (p q) ( p q)
p q p q pq p q (p q) (p q)
T T F F T F F
T F F T F T F
F T T F F T F
F F T T F T F
All the truth values in the last column are F. Hence it is contradiction.
Ex. 4) Prove that i) p → q p q ii) p → q (p → q) (q → p)
I II III IV V VI VII VIII
p q p p → q q → p p→q p → q p→qq→p
T T F T T T T T
T F F F T F F F
F T T T F F T F
F F T T T T T T
Columns (IV, VII) and (VI, VIII) are identical.
p → q p q and p q (p → q) (q → p) are proved.
1.3 Quantifiers, Quantified Statements, Duals, Negation of Compound Statements, Converse,
Inverse And Contra positive of Implication.
Quantifiers : A sentence that contains one or more variables is called an open sentence.
An open sentence becomes true or false statement when we replace the variables by some
specific values form a given set. The phrases that quantify the variables in open sentences
are called quantifiers.
, There are two types of quantifiers.
(i) Universal quantifiers : The quantifiers 'for all' or 'for every' is called universal
quantifiers and is denoted by .
(ii) Existential quantifier : The quantifiers' for some' or' for one' or 'for each' or 'there exist
at least one' or simply' there exists' is called existential quantifier and is denoted by .
Quantified statement : An open sentence with a quantifier becomes a statement in logic.
Such statement is called a quantified statement. i.e. statements involving quantifiers are
called quantified statements.
Remarks :
(i) Every quantified statement corresponds to a collection and a condition.
(ii) A statements quantified by universal quantifier for all '' is true if all objects in the
collection satisfy the condition and is false if at least one object in the collection does not
satisfy the condition.
(iii) A statement quantified by existential quantifiers '' is true if at least one object in the
collection satisfies the condition and is false if no object in the collection satisfies the
condition.
Ex. 1) If A = {1, 2, 3, 4, 5, 6, 7}, determine the truth value of the following.
i) x A such that x − 4 = 3 ii) x A , x+1>3
Solution : i) For x = 7, x – 4 = 7 – 4 = 3
x = 7 satisfies the equation x − 7 = 3
The given statement is true and its truth value is T.
ii) For x = 1, x + 1 = 1 + 1 = 2 which is not greater than or equal to 3.
For x = 1 x2 + 1 > 3 is not true.
The truth value of given statement is F.
Duals : Two compound statements S1 and S2, are said to be duals of each other if any
one of them can be obtained from the other by replacing conjunction () by disjunction
() and vice versa. The connectives and are called duals of each other. e.g. The
duals of p q is p q.
Remarks :
(i) If a compound statements 'S' contains tautology then is denoted by 't' and if it
contains contradiction then it is denoted by 'f'.
(ii) Two statements contain logical connectives like , and letters 't' and 'c' then
they are said to be dual of each other if one of them is obtained from other by
interchanging with and and 't' with 'c'. eg. The dual of ‘t p' is ‘c p’ and
the dual of ‘t p' is ‘c p’
Standard - XII : Subject - Mathematics and Statistics (Arts and Science) Part - I : 4
(Quick Guidance CHEAT SHEET)
Note – Questions given here are indicative and typical. However, in exams, only
these kind of questions will be asked
1. Mathematical Logic
1.1 Statement :
A statement is a declarative (assertive) sentence which is either true or false, but not both
simultaneously. Statements are denoted by p, q, r, .....
Truth value of a statement :
Each statement is either true or false. If a statement is true then its truth value is 'T' and
if the statement is false then its truth value is F.
Following sentences are statements.
i) 5 2 = 11
ii) Every triangle has three sides.
iii) Mumbai is the capital of Maharashtra.
Following sentences are not statements.
i) Please, give your Pen.
ii) What is your name?
iii) Open the window.
Note : Interrogative, exclamatory, command, order, request, suggestion are not statements.
Let us learn these statements :
i) For x = 6 it is true but for other than 6 it is not true. Here, we cannot determine the truth
value.
ii) For ‘It is black in colour.’ the truth value varies from person to person. In the above
sentences, the truth value depends upon the situation. Such sentences are called as open
sentences. Open sentence is not a statement.
Logical connectives, simple and compound statements :
The words or phrases which are used to connect two statements are called logical
connectives.
We will study the connectives 'and', 'or', 'if ..... then', 'if and only if ', 'not''.
Simple and Compound Statements : A statement which cannot be split further into two
or more statements is called a simple statement. If a statement is the combination of two
or more simple statements, then it is called a compound statement.
"3 is a prime and 4 is an even number", is a compound statement.
"3 and 5 are twin primes", is a simple statement.
,1.2 Statement Pattern, Logical Equivalence, Tautology, Contradiction, Contingency.
1) Statement Pattern : Letters used to denote statements are called statement letters.
Proper combination of statement letters and connectives is called a statement pattern.
Statement pattern is also called as a proposition. p → q, p q, p q are statement
patterns. p and q are their prime components.
A table which shows the possible truth values of a statement pattern obtained by
considering all possible combinations of truth values of its prime components is called
the truth table of the statement pattern.
2) Logical Equivalence :
Two statement patterns are said to be equivalent if their truth tables are identical. If
statement patterns A and B are equivalent, we write it as A B.
3) Tautology, Contradiction and Contingency :
⚫ Tautology : A statement pattern whose truth value is true for all possible combinations
of truth values of its prime components is called a tautology. We denote tautology
by t.
Ex. Statement pattern p p is a tautology.
⚫ Contradiction : A statement pattern whose truth value is false for all possible
combinations of truth values of its prime components is called a contradiction. We
denote contradiction by c.
Ex. Statement pattern p p is a contradiction.
⚫ Contingency : A statement pattern which is neither a tautology nor a contradiction
is called a contingency.
Ex. p q is a contingency.
Ex. 1) [(p q) r] (p (q r)]
p q r p q (p q) r q r p (q r) [(p q) r] [p (q r)]
T T T T T T T T
T T F T T T T T
T F T T T T T T
T F F T T F T T
F T T T T T T T
F T F T T T T T
F F T F T T T T
F F F F F F F T
All the truth values in the last column are T, hence it is tautology.
Standard - XII : Subject - Mathematics and Statistics (Arts and Science) Part - I : 2
,Ex. 2) [p p → q)] → q
p q q p → q p ( p → q p (p → q) → q
T T F F F T
T F T T T F
F T F T F T
F F T T F T
Truth values in the last column are not identical. Hence it is contingency.
Ex. 3) (p q) ( p q)
p q p q pq p q (p q) (p q)
T T F F T F F
T F F T F T F
F T T F F T F
F F T T F T F
All the truth values in the last column are F. Hence it is contradiction.
Ex. 4) Prove that i) p → q p q ii) p → q (p → q) (q → p)
I II III IV V VI VII VIII
p q p p → q q → p p→q p → q p→qq→p
T T F T T T T T
T F F F T F F F
F T T T F F T F
F F T T T T T T
Columns (IV, VII) and (VI, VIII) are identical.
p → q p q and p q (p → q) (q → p) are proved.
1.3 Quantifiers, Quantified Statements, Duals, Negation of Compound Statements, Converse,
Inverse And Contra positive of Implication.
Quantifiers : A sentence that contains one or more variables is called an open sentence.
An open sentence becomes true or false statement when we replace the variables by some
specific values form a given set. The phrases that quantify the variables in open sentences
are called quantifiers.
, There are two types of quantifiers.
(i) Universal quantifiers : The quantifiers 'for all' or 'for every' is called universal
quantifiers and is denoted by .
(ii) Existential quantifier : The quantifiers' for some' or' for one' or 'for each' or 'there exist
at least one' or simply' there exists' is called existential quantifier and is denoted by .
Quantified statement : An open sentence with a quantifier becomes a statement in logic.
Such statement is called a quantified statement. i.e. statements involving quantifiers are
called quantified statements.
Remarks :
(i) Every quantified statement corresponds to a collection and a condition.
(ii) A statements quantified by universal quantifier for all '' is true if all objects in the
collection satisfy the condition and is false if at least one object in the collection does not
satisfy the condition.
(iii) A statement quantified by existential quantifiers '' is true if at least one object in the
collection satisfies the condition and is false if no object in the collection satisfies the
condition.
Ex. 1) If A = {1, 2, 3, 4, 5, 6, 7}, determine the truth value of the following.
i) x A such that x − 4 = 3 ii) x A , x+1>3
Solution : i) For x = 7, x – 4 = 7 – 4 = 3
x = 7 satisfies the equation x − 7 = 3
The given statement is true and its truth value is T.
ii) For x = 1, x + 1 = 1 + 1 = 2 which is not greater than or equal to 3.
For x = 1 x2 + 1 > 3 is not true.
The truth value of given statement is F.
Duals : Two compound statements S1 and S2, are said to be duals of each other if any
one of them can be obtained from the other by replacing conjunction () by disjunction
() and vice versa. The connectives and are called duals of each other. e.g. The
duals of p q is p q.
Remarks :
(i) If a compound statements 'S' contains tautology then is denoted by 't' and if it
contains contradiction then it is denoted by 'f'.
(ii) Two statements contain logical connectives like , and letters 't' and 'c' then
they are said to be dual of each other if one of them is obtained from other by
interchanging with and and 't' with 'c'. eg. The dual of ‘t p' is ‘c p’ and
the dual of ‘t p' is ‘c p’
Standard - XII : Subject - Mathematics and Statistics (Arts and Science) Part - I : 4
