Q1.
Give three examples of sentences which are not statements. Give reason for the
answers.
Q2.
Write the negation of each of the following:
(i)All the students of this school live in the hostel.
(ii)
.
(iii)Ram is an intelligent boy.
Q3.
Let
. Determine the truth value of each of the following:
(i)
.
(ii)
Q4.
.
Write down the truth set of each of the following open sentences:
(i)
(ii)
Q5.
Use quantifiers to convert each of the following open sentences defined on
true statement:
, into
(i)
(ii)
Q6.
What is a compound statement? For each of the following compound statements,
first identify the connecting words and then break it into component statements
. Also write the truth value of each:
(i) is a rational number and
(ii)
(iii)
is an irrational number.
.
are the
factors of the quadratic equation
.
(iv)Two co-planar lines are parallel or intersect at a point.
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Q7.
State whether the “Or” used in the following statements is „exclusive‟ or „inclusive‟.
Give reasons for your answer.
(i)All integers are positive or negative.
(ii)A student who has taken Biology or Chemistry can apply for M.Sc. (Microbiology).
(iii)Two lines are parallel or they intersect at a point.
(iv)The school is closed if it is a Sunday or a holiday.
Q8.
Identify the quantifiers and write the negation of the following statements:
(i)There exists a number sum of whose factors is equal to the product of factors.
(ii)For every real number ,
is less than
.
(iii)There exists a roll number for each student of the class.
Q9.
Write the contrapositive of the following statements:
(i)If
(ii)If
, then their corresponding sides are equal.
, then
is divisible by
(iii)If quadrilateral
Q10.
is a parallelogram, then its opposite sides are parallel.
Write the converse of the following statements:
(i)If
is isosceles, then the base angles
are equal.
(ii)If
is right angled at
.
(iii)If three points
Q11.
.
, then
are collinear, then the area of
is equal to zero.
Write the contradiction of the following statements:
(i)If quadrilateral
(ii)If a function
is a rectangle, then diagonals are equal.
is continuous, then
(iii)If the discriminant of
imaginary roots.
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is differentiable.
is negative, then the equation has
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Q12.
Give the truth table for
following implications:
(i)If
(ii)If
(iii)If
Q13.
, then
is a multiple of .
is a multiple of , then
, then
is an odd number.
.
Write the truth value of each of the following biconditional statements:
(i)
if and only if
(ii) is odd if and only if
(iii)
Q14.
, also write down the truth value of each of the
.
is odd.
if and only if
.
Write the negation of the following:
(i)A triangle is equilateral if and only if it is equiangular.
(ii)
if and only if
.
(iii)He swims if and only if the water is clean.
Q15.
What is the negation of
? Write the negation of the following implications:
(i)Rashmi is tall, and therefore, she is slim.
(ii)If it rains, the humidity increases.
(iii)If the home work is no completed on time, the teacher will be angry.
Q16.
Two statements (a) and (b) are given below. Identify the statements given below
them as their contrapositive, converse and contradiction:
(a)If it rains, then I stay at home.
(i)If I stay at home, then it does not rain.
(ii) If I do not stay at home, then it does not rain.
(iii) If I stay at home, then it rains.
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(b)
.
(i)
.
(ii)
.
(iii)If
is not equiangular to
, then
Q17.
Prove that
Q18.
Rewrite the following in the form „if
is not congruent to
.
.
then ‟ or in the form „ if and only if
‟.
(i)If a quadrilateral is equiangular, then it is a rectangle and if a quadrilateral is a
rectangle, then it is equiangular.
(ii)There is a traffic jam whenever it rains.
(iii)For you to get an A grade, it is necessary and sufficient that you do your
homework regularly.
(iv)A number is even whenever it is divisible by two.
Q19.
By giving counter examples, show that the following statements are not true:
(i)If all the angles of a triangle are equal, it is an obtuse angled triangle.
(ii)The equation
(iii)If
Q20.
does not have a root lying between
.
is an odd integer, then it is prime.
Prove the following:
(i)
(ii)
.
.
Answers
A1.
Hint: Write sentences which are either true or false but not both.
A2.
(i)There is at least one student who does not live in the hostel.
(ii)
.
(iii)Ram is not an intelligent boy.
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A3.
(i) F (ii) F
A4.
(i)
A5.
(i)
(ii)
.
, such that
is a true statement, since
satisfies
.
(ii)
positive.
A6.
is a true statement, since the square of every natural number is
(i)‟and‟
is a rational number.
is an irrational number. (F)
(ii) „or‟
. (T)
(iii) „and‟
is a factor of the quadratic equation
is a factor of the quadratic equation
(iv) „or‟
.
. (T)
Two coplanar lines are parallel.
Two coplanar lines intersect at a point. (T)
A7.
(i)Exclusive; integers cannot be positive and negative simultaneously.
(ii)Inclusive; Students who have taken one of these subjects or both of these
subjects can apply for M.Sc. (Microbiology).
(iii) Exclusive; It is not possible for two lines to be parallel and intersect together.
(iv) Inclusive; School is closed on Sundays as well as on holidays.
A8.
(i)There exists; There does not exist a number, sum of whose factors is equal to the
product of factors.
(ii)For every; There exist at least one real number
.
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such that
is not less than
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(iii) There exists; There exists at least one student in the class who does not have
any roll number.
A9.
(i)If the sides of
(ii)If
are not equal, then
is not divisible by
, then
.
(iii)If the opposite sides of the quadrilateral
a parallelogram.
A10.
(i)If the base angles
of
(i)If the diagonals of quadrilateral
(ii)If the function
is not
is isosceles.
is right angled at
are collinear.
.(iii)If the area of
are not equal, then it is a rectangle.
is not differentiable, then
(iii)If the equation
discriminant is negative.
A12.
are not parallel, then
are equal, then
(ii)If in
,
, then the
is equal to zero, then the points
A11.
.
is continuous.
does not have imaginary roots, then its
Truth table for
(i)F (ii) T (iii) T.
A13.
(i)F (ii) F (iii) T
A14.
(i)There exists either an equilateral triangle which is not equiangular or an
equiangular triangle which is not equilateral.
(ii)Either
and
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or
and
.
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(iii)Either he swims and the water is not clean or the water is clean and he does not
swim.
A15. Negation of
is
.
(i)Rashmi is tall and she is not slim.
(ii)It rains and the humidity does not increase.
(iii)The home work is not completed on time and the teacher will not be angry.
A16.
(a) (i) Contradiction (ii) Contrapositive (iii) Converse
(b) (i) Converse
(ii) Contradiction
(iii) Contrapositive
A17.
Hint:
A18.
(i) A quadrilateral is equiangular if and only if it is a rectangle.
(ii) If it rains, then there is a traffic jam.
(iii) You get an A grade if and only if you do your home work regularly.
(iv) If a number is divisible by , then it is even.
A19.
(i)Each of the angles of the triangle is acute
(ii)Two roots are
(iii)
A20.
, the root
is the counter example.
lying between
is a counter example.
is a counter example.
Hint: (i) Show that the truth table of
(ii)Show the truth table of
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and
are identical.
. This must be same as the column of
.
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