Exercise 14.1
Q.1: State whether the following sentences are statements or not, and justify your answers.
(a) A month has 35 days.
(b) Mathematics is very tough
(c) Addition of two numbers such as 5 & 7 is larger than 10.
(d) The resultant of a square of a number is always an even number.
(e) The arms of a quadrilateral are having equal length.
(f) Answer the following questions.
(g) The multiplication result of two numbers such as 8 and (-1) is 8
(h) The interior angles summed up together results in 1800 in a triangle.
(i) Yesterday was a cloudy day.
(j) The numbers which are real are always complex numbers.
Answer:: Based on the formulae given in the
(a) The maximum number of days in a month is 31.Hence, the statement is incorrect. Therefore, this isn’t a proved statement.
(b) Mathematics can be tough for someone and it can be easy for someone too. Hence it is a direct violation to the above question. Hence, this isn’t a statement.
(c) The sum of two numbers such as 5 and 7 is 12 which is larger than 10. Hence, the above-mentioned sentence is true. Therefore, this is a proved statement.
(d) Square of all numbers does not give even number as a result. Such as the square of a even number that is 4 is 16 which is a even number, but the square of a odd number such as 9 is 81 which is a odd number. Hence, the above mentioned statement is not fully true. Therefore, this isn’t a proved statement.
(e) The above sentence is true only in certain cases. In case of square, all the sides are having equal length whereas in case of a rectangle all the sides are not equal except the opposite sides are equal in length. Hence, above the statement is not true. Therefore, this isn’t a proved statement.
(f) Above mentioned sentence is a direct order given to someone in order to answer the upcoming questions. Therefore, this isn’t a proved statement.
(g) The multiplication of 8 and -1 is (–8). Therefore, the above mentioned sentence is incorrect. Hence, this is a proved statement.
(h) This sentence is correct because the addition of all interior angles of a triangle is 1800. Hence, this is a proved statement.
(i) The day which is being referred to in the above mentioned sentence is not at all clear. Therefore, this isn’t a proved statement.
(j) All real numbers can be written in a format such as (1) + 0(i) Therefore, the given sentence is always correct. Hence, this is a proved statement.
Q.2: Give 3 examples of each sentence which are not statements. Give justified reasons for the answers.
Answer:: Based on the formulae given in the
The three examples of sentences, which are not statements, are as follows.
(i) Today is a cloudy day.
The day which is being referred to in the above mentioned sentence is not at all clear. Therefore, it is not a statement.
(ii) Mathematics is very tough
Mathematics can be tough for someone and it can be easy for someone too. Hence it is a direct violation to the above question. Therefore, it is not a statement.
(iii) Where are you going?
The above sentence is a question which contains ‘ you ‘ , and it is not at all clear from the above question who is being referred to . Therefore, it is not a statement.
Exercise 14.2
Q.1: Write the opposite of the below mentioned statements:
(a). New Delhi is the capital of India.
(b). \(\sqrt{3+1}\) is a complex number.
(c). All quadrilaterals are not squares.
(d). 9 is lesser than 7
(e). The square of every natural number is a even number
Answer:: Based on the formulae given in the
(a) New Delhi is not the capital of India
(b) \(\sqrt{3+1}\) is not a complex number.
(c) All quadrilaterals are squares.
(d) 9 is larger than 7
(e) The square of every natural number is not a even number.
Q.2: State whether the following statements are opposite to each other or not.
(a). The number 2 is not an even number
The number 2 is not an odd number.
(b). The number 2 is an even number.
The number 2 is an odd number.
Answer:: Based on the formulae given in the
(a). The opposite of the Statement I is “The number 2 is even number “. The above statement is same as Statement II. The reason is if a number is not odd number then it is even number.
(b). The opposite of Statement I is “The number 2 is not an even number”. The above statement is same as Statement II. The reason is if a number is not odd number then it is even number.
Q.3: Find out the component sentences from the below mentioned compound sentences, and determine whether they are true/ false.
(a). Number 5 is odd or it is a prime number.
(b). All integers are positive and negative.
(c). 1000 is divisible by 9 or 10
Answer:: Based on the formulae given in the
(a).
(i). Number 5 is odd
(ii). Number 5 is a prime number.
In this case both the statements are true.
(b).
(i). All integers are positive
(ii). All integers are negative
In this case both the statements are false.
(c).
(i). 1000 is divisible by 9
(ii). 1000 is divisible by 10
In this case only statement (ii) is true.
Exercise 14.3
Q.1:
(i). Every real number is not complex number and every rational number is a real number.
(ii). Square of any integer is negative or positive.
(iii). The sand easily heats up due to the sun but does not cool down easily at night
(iv). The roots for the equation x + 10 = 3x2 are x = 3 and x = 2
Answer:: Based on the formulae given in the
(i) Here ‘and’ is a connecting word. Here component statements will be:
a: Every real number is not complex.
b: Every rational number are real.
(ii) Here, ‘or’ is a connecting word. Here component statements will be:
a: Square of any integer is negative.
b: Square of any integer is positive.
(iii) Here ‘but’ is connecting word. Here component statements will be:
a: The sand heats up easily due to sun.
b: The sand dose not cool down easily at night.
(iv) Here ‘and’ is the connecting word. Here component statements will be:
a: The roots for the equation x + 10 = 3x2 are x = 3
b: The roots for the equation x + 10 = 3x2 are x = 2
Q.2: Write negation for the statements after identifying the quantifier for the statements
(i). There exits one number that is equal to the square of the number
(ii). For every number that is real ‘x’, x < x + 1
(iv). There exist one capital for each state of India.
Answer:: Based on the formulae given in the
(i) The quantifier will be “There exist” and negation for the statement is:
There doesn’t exist any number that is equal to the square of the number.
(ii) The quantifier will be “ For every” and negation for the statement is:
There exist a number x that is not less than x + 1
(iii) The quantifier will be “There exist” and negation for the statement is:
There exist one state that has no capital.
Q.3: Check if the following statements are negation for each other. Justify your answer
(i). y + x = x + y is true for real numbers x and y
(ii). There exist real numbers x, y such that y + x = x + y
Answer:: Based on the formulae given in the
The negation for statement (i) will be:
There exist real numbers x, y such that y + x ≠ x + y, which is not statement (ii).
So the statements are not negation for each other.
Q.4: State if the “Or” in the statements is inclusive or exclusive. Justify the answer
(i) Moon sets or sun rises
(ii) You must have ration card or passport for applying a driving license.
(iii) Integers are negative or positive
Answer:: Based on the formulae given in the
(i) “or” is exclusive as it is impossible for moon to set and sun to rise together.
(ii) “or” is inclusive as one can have both passport and ration card for applying a driving license.
(iii) “or” is exclusive as integers can’t be negative and positive.
Exercise – 14.4
Q.1: Rewrite the statements with ‘if & then’ in 5 different ways but the sentence should convey the meaning as before.
A natural number is odd implies that its square is odd.
Answer:: Based on the formulae given in the
(a). A If the square of the natural number is said to be odd then the number is odd too.
(b). If the square of a number is not even then the natural number which is squared is also odd.
(c). It is mandatory that the square of a number to be not even (odd) in order to have that natural number as odd.
(d). If the natural number is not odd then the square of that number is also not odd that is even.
(e). In order to have the square of a number (natural number) which is odd, it is sufficient that the number (natural number) is also odd.
Q.2: Rewrite the following sentences as the converse/contrapositive of the followings:
(a). A quadrilateral is said to be parallelogram if the diagonals bisect each other.
(b). y is an odd number that is y is divisible by 3
(c). If 2 lines do not intersect in the same plane, then they are said to be parallel.
(d). If something is having a low temperature then it implies that is cold
(e). If you are not able to deduct the reason, then you will not be able to comprehend geometry.
Answer:: Based on the formulae given in the
(a) Contrapositive – If the diagonals of a quadrilateral do not bisect each other, then the quadrilateral is not a parallelogram.
Converse – If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
(b) Contrapositive – If y is not divisible by 3 then it is not an odd number.
Converse – If y is divisible by 3 , then it is an odd number.
(c) Contrapositive – If 2 lines intersect in the same plane, then they are not parallel.
Converse – If 2 lines do not intersect in the same plane, then they are parallel.
(d) Contrapositive – If something does not have low temperature, then it implies that it is not cold
Converse – If something is having at a low temperature, then it implies that it is cold.
(e) Contrapositive – If you know how to reason deductively, then you can comprehend geometry.
Converse – If you do not know how to reason deductively, then you cannot comprehend geometry.
Q.3: Rewrite the following sentences with “if- then”:
(a). You have visited Qutub Minar implies that you live in Delhi
(b). You will pass the exam if you study hard.
(c). In order to get A+ in the class test, you have to do all the problems of that chapter.
(d). Parallel lines do not intersect each other in the same plane
Answer:: Based on the formulae given in the
(a) If you have visited Qutub Minar then you live in Delhi.
(b) If you study hard then you will pass the exam
(c) If you want to get A+ in the class test then you have to do all the problems of the book.
(d) If two lines are parallel then they do not intersect each other in the same plane.
Q.4: Identify the contrapositive/converse from the following sentences:
(i) If you live in Agra, then you have visited Taj Mahal
(a) If you have not visited Taj Mahal then you do not live in Agra.
(b) If you have visited Taj Mahal then you live in Agra.
(ii) If the diagonals of the quadrilateral bisect each other then that quadrilateral is a parallelogram.
(a) A quadrilateral is not said to be a parallelogram if the diagonals of a quadrilateral do not bisect each other.
(b) A quadrilateral is said to be a parallelogram if the diagonals of a quadrilateral bisect each other.
Answer:: Based on the formulae given in the
(i).
(a) Statement I is the contrapositive of the above mentioned statement
(b) Statement II is the converse of the above mentioned statement
(ii).
(a) Statement I is the contrapositive of the above mentioned statement
(b) Statement II is the converse of the above mentioned statement
Exercise 14.5
Q.1: Prove that p: “If a is real such that a3+ 4a = 0, then a is 0″ is true
(i). by direct method
(ii). by method of contradiction
(iii). by method of contra positive
Answer:: Based on the formulae given in the
p: “if a is real such that a3+ 4a = 0, then a is 0”
Let q: a is real such that a3+ 4a = 0 r, a is 0.
(i). To show that “p” is true, we take that “q” must be true and then prove that “r” is true.
Therefore, assume statement “q” be true.
a3+ 4a= 0 a (a2 + 4) = 0
=> a = 0 or a2 + 4 = 0
However, since a is real, so it is O.
Hence, “r” is true.
Thus, the statement is true.
(ii). To show “p” is true using contradiction, we take that “p” isn’t true.
Let x be a real number such that a3+ 4a = 0 and let x is not 0.
Therefore, a3+ 4a = 0 x (a2 + 4) = 0 a = 0 or a2 + 4 = 0 a = 0 or a2= — 4
However, ‘a’ is real. Thus, a = 0, which is contradiction as we assume that a is not 0.
Hence, the statement “p” is true.
(iii). To prove “p” to be true by using contrapositive method, let r is false and show that q is false.
Here, “r” is false states that its requirement of the negation for statement r.
This obtains the following statement.
~ r. x is not 0.
It is seen (a2 + 4) cannot be negative so it will be positive.
a ≠ 0 states that product of a positive number with “a” is not zero.
Let us assume the product of a with (a2 + 4).
a (a2+ 4) ≠ 0
a3 + 4a ≠ 0
This proves that “q” isn’t true.
Hence, it is proved that ~ r => ~ q
Therefore, the statement “p” is true.
Q.2: Prove the statement “For real numbers b and a, b2 = a2 implies that b = a” isn’t true. Give a counter example.
Answer:: Based on the formulae given in the
Using “if- then” the given statement may be written as follows.
If b and a are two real numbers and b2 = a2, then b = a
Let p: b and a are two real numbers and b2 = a2
q: b = a
To prove: given statement is false. For this we need to prove that p. then ~q. To prove this, we need two numbers b and a with b2 = a2 such that b ≠ a (the numbers must be real numbers)
Let b = (-1) and a = 1 b2 = (-1)2 = 1 and a2 = (1)2 = 1
Therefore b2 = a2
However, b ≠ a
Thus, it is proved that given statement is false.
Q3: By using contrapositive method prove the following statement is true
p: if a is an integer and a2 is even, then a is even.
Answer:: Based on the formulae given in the
p: if a is an integer and a2 is even, then a is even.
Assume q: a is an integer with “a2” even. r. a is even
To show: By using contrapositive method p is true, q is false.
r is false [we assume]
let a is odd number
To show that ‘q’ is false, we have to prove that ‘a’ isn’t an integer or a2 isn’t even
‘a’ isn’t even implies that a2 is not even
Therefore, ‘q’ is false. Thus, given statement i.e p is true.