**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 180 ^{0} in a triangle.**

**(i) Yesterday was a cloudy day. **

**(j) The numbers which are real are always complex numbers. **

** **

**Sol:**

**(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 180^{0}. **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.**

** **

**Sol: **

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**

** **

**Sol:**

**(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.**

** **

**Sol:**

**(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**

** **

**Sol:**

**(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 = 3x ^{2} are x = 3 and x = 2**

** **

**Sol:**

**(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 = 3x ^{2} are x = 3**

**b: The roots for the equation x + 10 = 3x ^{2} 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.**

**Sol:**

**(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**

** **

**Sol:**

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**

** **

**Sol:**

**(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. **

** **

**Sol:**

**(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.**

** **

** Sol:**

**(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 **

** **

**Sol:**

**(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.

** **

**Sol:**

**(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 a ^{3}+ 4a = 0, then a is 0″ is true **

**(i). by direct method**

**(ii). by method of contradiction**

**(iii). by method of contra positive**

**Sol:**

**p: “if a is real such that a ^{3}+ 4a = 0, then a is 0”**

**Let q: a is real such that a ^{3}+ 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.

a^{3}+ 4a= 0 a (a^{2} + 4) = 0

=> a = 0 or a^{2 }+ 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 a^{3}+ 4a = 0 and let x is not 0.

Therefore, a^{3}+ 4a = 0 x (a^{2} + 4) = 0 a = 0 or a^{2} + 4 = 0 a = 0 or a^{2}= — 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 (a^{2} + 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 (a^{2} + 4).

a (a^{2}+ 4) ≠ 0

a^{3} + 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, b ^{2} = a^{2} implies that b = a” isn’t true. Give a counter example.**

**Sol:**

Using “if- then” the given statement may be written as follows.

If b and a are two real numbers and b^{2} = a^{2}, then b = a

Let p: b and a are two real numbers and b^{2} = a^{2}

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 b^{2} = a^{2} such that b ≠ a (the numbers must be real numbers)

Let b = (-1) and a = 1 b^{2} = (-1)^{2} = 1 and a^{2} = (1)^{2} = 1

Therefore b^{2} = a^{2}

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 a ^{2} is even, then a is even.**

**Sol:**

**p:** if a is an integer and a^{2} is even, then a is even.

**Assume q:** a is an integer with “a^{2}” 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 a^{2} isn’t even

‘a’ isn’t even implies that a^{2} is not even

**Therefore, ‘q’ is false. Thus, given statement i.e p is true.**