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Let us assume that ‘q’ and ‘r’ be the statements given

q: x is an integer and x² is odd.

r: x is an odd integer.

The given statement can be written as:

p: if q, then r.

Let r be false. Then,

x is not an odd integer, then x is an even integer

x = (2n) for some integer n

x² = 4n²

x² is an even integer

Thus, q is False

Therefore, r is false and q is false

Hence, p: “ if q, then r” is a true statement.

(i) A given number is a multiple of 6, it implies that it is a multiple of 3 as well.

(ii) For a given number to be a multiple of 6, it is necessary that it is a multiple of 3.

(iii) A given number is a multiple of 6 only if it is a multiple of 3.

(iv) If the given number in not a multiple of 3, then it is not a multiple of 6.

(v) For a given number to be a multiple of 3, it is sufficient that the number is multiple of 6.

Let us Assume that p and q be the statements given by 

p: x and y are odd integers.

q: x + y is an even integer 

since the given statement can be written as : 

if p, then q. 

Let p be true . then, 

x and y are odd integers 

x = 2m+1, y = 2n+1 for some integers m, n 

x + y = (2m+1)+(2n+1) 

x + y = (2m+2n+2) 

x + y = 2(m+n+1) 

x + y is an integer q is true. 

Therefore, p is true ⇒ q is true 

Hence, “if p, then q “ is a true statement.”

(i) Statement: 100 is a multiple of 4 and 5. 

Here, we know that 100 is a multiple of 4 as well as 5. So, statement p is true. 

Hence, The statement is true, Therefore The statement “p” is a valid statement. 

(ii) Statement: 125 is a multiple of 5 and 7 

Since 125 is a multiple of 5, but it is not a multiple of 7. So, The statement “q” is not a true statement. 

Hence, The statement “q” id not a valid statement. 

(iii) Statement: 60 is a multiple of 3 or 5. 

Here, we know that 60 is a multiple of 3 as well as 5. So, statement r is true. 

Hence, The statement is true. Therefore The statement “r” is a valid statement.

(i) You watch television if and only if your mind is free.

(ii) A quadrilateral is a rectangle if and only if it is equiangular.

(iii) You get an A grade if and only if you do all the homework regularly.

(iv) A tumbler is half empty if and only if it is half full.

(i) p: 100 is a multiple of 4 and 5.

We know that 100 is a multiple of 4 as well as 5. So, the given statement is true.

Hence, the statement is true.

(ii) q: 125 is a multiple of 5 and 7

We know that 125 is a multiple of 5 and not a multiple of 7. So, the given statement is false.

Hence, the statement is false.

(iii) r: 60 is a multiple of 3 or 5.

We know that 60 is a multiple of 3 as well as 5. So, the given statement is true.

Hence, the statement is true.

(i) If Mohan is a poet, then he is poor.

Contrapositive: If Mohan is not poor, then he is not a poet.

(ii) Only if Max studies will he pass the test.

Contrapositive: If Max does not study, then he will not pass the test.

(iii) If she works, she will earn money.

Contrapositive: If she does not earn money, then she does not work.

(iv) If it snows, then they do not drive the car.

Contrapositive: If then they do not drive the car, then there is no snow.

(v) It never rains when it is cold.

Contrapositive: If it rains, then it is not cold.

(vi) If Ravish skis, then it snowed.

Contrapositive: If it did not snow, then Ravish will not ski.

(vii) If x is less than zero, then x is not positive.

Contrapositive: If x is positive, then x is not less than zero.

(viii) If he has courage he will win.

Contrapositive: If he does not win, then he does not have courage.

(ix) It is necessary to be strong in order to be a sailor.

Contrapositive: If he is not strong, then he is not a sailor

(x) Only if he does not tire will he win.

Contrapositive: If he tries, then he will not win.

(xi) If x is an integer and x² is odd, then x is odd.

Contrapositive: If x is even, then x² is even.

(i) Statement: If Mohan is a poet, then he is poor. 

Contrapositive : If Mohan is not poor, then he is not a poet. 

(ii) Statement: Only if Max studies will he pass the test. 

Contrapositive : If Max does not study, then he will not pass the test. 

(iii) Statement: If she works, she will earn money. 

Contrapositive : If she does not earn money, then she does not work. 

(iv) Statement: If it snows, then they do not drive the car.

Contrapositive : If then they do not drive the car, then there is no snow. 

(v) Statement: It never rains when it is cold. 

Contrapositive : If it rains, then it is not cold. 

(vi) Statement: If Ravish skis, then it snowed. 

Contrapositive : If it did not snow, then Ravish will not ski. 

(vii) Statement: If x is less than zero, then x is not positive. 

Contrapositive : If x is positive, then x is not less than zero. 

(viii) Statement: If he has courage he will win. 

Contrapositive : If he does not win, then he does not have courage. 

(ix) Statement: It is necessary to be strong in order to be a sailor. 

Contrapositive : If he is not strong, then he is not a sailor 

(x) Statement: Only if he does not tire will he win. 

Contrapositive : If he tries, then he will not win. 

(xi) Statement: If x is an integer and x² is odd, then x is odd. 

Contrapositive : If x is even, then x² is even.

(a) Let p: Sun is not shining.

and q:Sky is filled with clouds.

Converse of the above statement p → q is q → p.

If sky is filled with clouds, then the Sun is not shining.

The given statement can be written in the form of “if-then” as follows.

If a and b are real numbers such that a² = b², then a = b.

Let p: a and b are real numbers such that a² = b².

q: a = b

The given statement has to be proved false. For this purpose, it has to be proved that if p, then ∼q. To show this, two real numbers, a and b, with a² = b² are required such that a ≠

b. Let a = 1 and b = –1 a² = (1)² = 1 and b² = (– 1)² = 1

∴ a² = b²

However, a ≠ b

Thus, it can be concluded that the given statement is false.