Mathematical Reasoning

Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.

So, the given sentence “A triangle has three sides” is true.

Hence, It is a true statement

Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.

So, The given sentence “0 is a complex number” is true. Because we can write it as a+ib, where imaginary part is 0 as, a+0i.

Hence, It is a true statement

Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.

The given sentence “sky is Red” is false.

Hence, It is a false statement

Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.

The given sentence “Every set is an infinite set” is False.

Hence, It is a false statement

Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.

So, The given Expression “15 + 8 > 23” is false. As the result of L.H.S will always equal to the result of R.H.S

Hence, It is a false statement

Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.

Here, y + 9 = 7 will be true for some value and it will be false for some value .

Like, at y= -2, the given expression is true and

Y=1 or any other value, the expression is false.

Hence, It is not a statement

Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.

Here, The given sentence “Where is your bag” is a question.

Hence, It is not a statement

Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.

So, The given sentence “Every square is a rectangle.” is true.

Hence, It is a true statement

Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.

By the properties of quadrilateral “Sum of opposite angles of a cyclic quadrilateral is 180°.”

So, The given sentence is true.

Hence, It is a statement

Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.

According to the laws of trigonometry, sin²x + cos²x = 1

So, The given expression is False.

Hence, It is a false statement

Concept Used:A compound statement is a combination of two statements (Components).

So, The components of the given statement “Number 7 is prime and odd” are,

p:Number 7 is prime.

q: Numeer 7 is odd.

Concept Used:A compound statement is a combination of two statements (Components).

So, the components of the given statement “Chennai is in India and is the capital of Tamil Nadu” are,

p: Chennai is in India.

q: Chennai is the capital of Tamil Nadu

Concept Used: A compound statement is a combination of two statements (Components).

So, The components of the given statement “The number 100 is divisible by 3, 11 and 5”.

p: 100 is divisible by 3.

q: 100 is divisible by 11.

r: 100 is divisible by 5.

Concept Used: A compound statement is a combination of two statements (Components).

So, The components of the given statement “Chandigarh is the capital of Haryana and U.P.

” are,

p: Chandigarh is the capital of Haryana and U.P

q: Chandigarh is the capital of U.P

Concept Used: A compound statement is a combination of two statements (Components).

So, The components of the given statement “ is a rational number or an irrational number,

p: is a rational number.

q: is an irrational

Concept Used: A compound statement is a combination of two statements (Components).

So, The components of the given statement “0 is less than every positive integer and every negative integer “ are

p: 0 is less than every positive integer

q: 0 is less than every negative integer.

Concept Used: A compound statement is a combination of two statements (Components).

So, The components of the given statement “ Plants use sunlight, water and carbon dioxide for photosynthesis” are

p: Plants use sunlight for photosynthesis

q: Plants use water for photosynthesis

r: Plants use carbon dioxide for photosynthesis

Concept Used: A compound statement is a combination of two statements (Components).

So, The components of the given statement Two lines in a plane either intersect at one point or they are parallel “ are

p: Two lines in a plane intersect at one point.

q: Two lines in a plane are parallel.

Concept Used: A compound statement is a combination of two statements (Components).

So, The components of the given statement “ A rectangle is a quadrilateral or a 5 - sided polygon” are

p: A rectangle is a quadrilateral

q: A rectangle is a 5 - sided polygon.

Concept Used: A compound statement is a combination of two statements (Components).

So, the components of the given statement “57 is divisible by 2 or 3” are

p: 57 is divisible by 2.

q: 57 is divisible by 3.

Now, The given compound statement is in the form of P V Q, that has truth value T whenever either P or Q or both will true.

Hence, The given statement is True.

Concept Used: A compound statement is a combination of two statements (Components).

So, The components of the given statement “24 is a multiple of 4 and 6” are

p: 24 is a multiple of 4.

q: 24 is a multiple of 6.

Now, Both the component p and q are true. As 24 is a multiple of both 4 and 6

Hence, The given statement is True.

Concept Used: A compound statement is a combination of two statements (Components).

So, The components of the given statement “All living things have two eyes and two legs” are

p: All living things have two eyes.

q: All living things have two legs

Now, The given compound statement is in the form of P ᴧ Q, that has truth value True

Only when, both the components will be true.

Here,

“All living things have two eyes” is False

“All living things have two legs” is False

Hence, The given statement is False.

Concept Used: A compound statement is a combination of two statements (Components).

So, The components of the given statement “2 is an even number and a prime number” are

p: 2 is an even number.

q: 2 is an prime number.

Now, The given compound statement is in the form of P ᴧ Q, that has truth value True

Only when, both the components will be true.

Here,

“: 2 is an even number” is True

“2 is an prime number” is True

Hence, The given statement is True.

Negation of statement p is "not p." The negation of p is symbolized by "~p." The truth value of ~p is the opposite of the truth value of p.

The negation of the statement is “The number 17 is not prime”.

Negation of statement p is "not p." The negation of p is symbolized by "~p." The truth value of ~p is the opposite of the truth value of p.

The negation of the statement is “2 + 7 ≠ 6”.

Negation of statement p is "not p." The negation of p is symbolized by "~p." The truth value of ~p is the opposite of the truth value of p.

The negation of the statement is “Violets are not blue”.

Negation of statement p is "not p." The negation of p is symbolized by "~p." The truth value of ~p is the opposite of the truth value of p.

The negation of the statement is “ is not a rational number. ”.

Negation of statement p is "not p." The negation of p is symbolized by "~p." The truth value of ~p is the opposite of the truth value of p.

The negation of the statement is “2 is a prime number”.

Negation of statement p is "not p." The negation of p is symbolized by "~p." The truth value of ~p is the opposite of the truth value of p.

The negation of the statement is “Every real number is not an irrational number”.

Negation of statement p is "not p." The negation of p is symbolized by "~p." The truth value of ~p is the opposite of the truth value of p.

The negation of the statement is “Cow does not have four legs”.

Negation of statement p is "not p." The negation of p is symbolized by "~p." The truth value of ~p is the opposite of the truth value of p.

The negation of the statement is “A leap year does not have 366 days”.

Negation of statement p is "not p." The negation of p is symbolized by "~p." The truth value of ~p is the opposite of the truth value of p.

The negation of the statement is “All similar triangle are not congruent”.

Negation of statement p is "not p." The negation of p is symbolized by "~p." The truth value of ~p is the opposite of the truth value of p.

The negation of the statement is “Area of a circle is not same as the perimeter of the circle.”.

(i)The given sentence is a compound statement in which components are

p:Rahul passed in Hindi

q:Rahul passed in English

Now, It can be represent in symbolic function as,

p ᴧ q: Rahul passed in Hindi and English.

(ii) The given sentence is a compound statement in which components are

p:x is an even integer

q:y is an even integer

Now, It can be represent in symbolic function as,

p ᴧ q: x and y are even integers.

(iii) The given sentence is a compound statement in which components are

p:2 is a factor of 12

q:3 is a factor of 12

r: 6 is a factor of 12

Now, It can be represent in symbolic function as,

p ᴧ q ᴧ r: 2, 3 and 6 are factors of 12.

(iv) The given sentence is a compound statement in which components are

p:x is an odd integer

q:x+1 is an odd integer

Now, It can be represent in symbolic function as,

p V q: Either x or x + 1 is an odd integer.

(v) The given sentence is a compound statement in which components are

p:A number is divisible by 2

q:A number is divisible by 3

Now, It can be represent in symbolic function as,

p V q: A number is either divisible by 2 or 3.

(vi) The given sentence is a compound statement in which components are

p: x = 2 is a root of 3x² – x – 10 = 0

q: x = 3 is a root of 3x² – x – 10 = 0

Now, It can be represent in symbolic function as,

p V q: Either x = 2 or x = 3 is a root of 3x² – x – 10 = 0

(vii) The given sentence is a compound statement in which components are

p: Hindi is the optional paper

q: English is the optional paper

Now, It can be represent in symbolic function as,

p ᴧ q: Either Hindi or English is optional paper.

(i) The given statement is compound statement then components are,

P:All rational numbers are real.

~p: All rational numbers are not real.

q: All rational numbers are complex.

~q: All rational numbers are not complex.

(p ᴧ q)= All rational numbers are real and complex.

~(p ᴧ q)=~p v ~q= All rational numbers are neither real nor complex.

(ii) The given statement is compound statement then components are,

P:All real numbers are rational.

~p: All real numbers are not rational.

q: All real numbers are irrational.

~q: All real numbers are not irrational.

(p ᴧ q)= All real numbers are rationals or irrationals.

~(p ᴧ q)=~p v ~q= All real numbers are neither rationals nor irrationals.

(iii) The given sentence is a compound statement in which components are

p: x = 2 is a root of Quadratic equation x² – 5x + 6 = 0.

~p: x = 2 is not a root of Quadratic equation x² – 5x + 6 = 0.

q: x = 3 is a root of Quadratic equation x² – 5x + 6 = 0.

~q: x = 3 is not a root of Quadratic equation x² – 5x + 6 = 0.

(p ᴧ q)= x = 2 and x = 3 are roots of the Quadratic equation x² – 5x + 6 = 0.

~(p ᴧ q)=~p v ~q= Neither x = 2 and nor x = 3 are roots of x² – 5x + 6 = 0

(iv) The given statement is compound statement then components are,

P:A triangle has 3 sides

~p: A triangle does not have 3 sides.

q: A triangle has 4 sides.

~q: A triangle does not have 4 side.

(p v q)= A triangle has either 3-sides or 4-sides.

~(p v q)=~p ᴧ ~q= A triangle has neither 3 sides nor 4 sides.

(v) The given statement is compound statement then components are,

P: 35 is a prime number

~p: 35 is not a prime number.

q: 35 is a composite number

~q: 35 is not a composite number.

(p v q)= 35 is a prime number or a composite number.

~(p v q)=~p ᴧ ~q= 35 is not a prime number and it is not a composite number.

(vi) The given statement is compound statement then components are,

P: All prime integers are even

~p: All prime integers are not even.

q: All prime integers are odd

~q: All prime integers are not odd.

(p v q)= All prime integers are either even or odd.

~(p v q)=~p ᴧ ~q= All prime integers are not even and not odd.

(vii) The given statement is compound statement then components are,

P: |x| is equal to x.

~p: |x| is not equal to x.

q: |x| is equal to –x.

~q: |x| is not equal to -x.

(p v q)= |x| is equal to either x or – x.

~(p v q)=~p ᴧ ~q= |x| is not equal to x and |x| is not equal to – x.

(viii) The given statement is compound statement then components are,

P: 6 is divisible by 2

~p: 6 is not divisible by 2

q: 6 is divisible by 3

~q: 6 is not divisible by 3.

(p ᴧ q)= 6 is divisible by 2 and 3.

~(p ᴧ q)=~p v ~q= 6 is neither divisible by 2 nor 3

(i) In the conditional statement, expression is

If p, then q

Now,

The given statement p and q are

p: The number is odd.

q: The square of odd number is odd.

Therefore,

“If the number is odd, then its square is odd number.

(ii) In the conditional statement, expression is

If p, then q

Now,

The given statement p and q are

p: Take the dinner

q: you will get sweet dish

Therefore,

“If take the dinner, then you will get sweet dish.

(iii) In the conditional statement, expression is

If p, then q

Now,

The given statement p and q are

p: You do not study

q: you will fail.

Therefore,

“If you do not study, then you will fail.”

(iv) In the conditional statement, expression is

If p, then q

Now,

In the given statement p and q are

p: An integer is divisible by 5

q: Unit digits of an integer are 0 or 5

Therefore,

“If an integer is divisible by 5, then its unit digits are 0 or 5.

(v) In the conditional statement, expression is

If p, then q

Now,

The given statement p and q are

p: Any number is prime,

q: square of number is not prime.

Therefore,

“If any number is prime, then its square is not prime”.

(vi) In the conditional statement, expression is

If p, then q

Now,

The given statement p and q are

p: a, b and c are in AP

q: 2b=a + c

Therefore,

“If a, b, c are in AP then 2b=a + c.

In the biconditional statement, we use if and only if.

p : The unit digit of an integer is zero.

q : It is divisible by 5.

Then,

Unit digit of an integer is zero if and only if it is divisible by 5.

In the biconditional statement, we use if and only if.

p : A natural number n is odd.

q : q : Natural number n is not divisible by 2.

Then,

A natural number is odd if and only if it is not divisible by 2.

In the biconditional statement, we use if and only if.

p : A triangle is an equilateral trinagle

q : All three sides of a triangle are equal.

Then,

A triangle is an equilateral triangle if and only if all three sides of triangle are equal.

(i) Definition of contrapositive: A conditional statement is logically equivalent to its contrapositive.

Contrapositive: If x≠3, then x ≠ y or y≠3

(ii) Definition of contrapositive: A conditional statement is logically equivalent to its contrapositive.

Contrapositive: If n is not an integer, then it is not a natural number.

(iii) Definition of contrapositive: A conditional statement is logically equivalent to its contrapositive.

Contrapositive: If the triangle is not equilateral, then all three sides of the triangle are not equal.

(iv) Definition of contrapositive: A conditional statement is logically equivalent to its contrapositive.

Contrapositive: if xy is not positive integer, then x or y is not negative integer.

(v) Definition of contrapositive: A conditional statement is logically equivalent to its contrapositive.

Contrapositive: If natural number ‘n’ is not divisible by 2 or 3, then n is not divisible by 6.

(vi) Definition of contrapositive: A conditional statement is logically equivalent to its contrapositive.

Contrapositive: The weather will not be cold, if it does not snow.

(vii) Definition of contrapositive: A conditional statement is logically equivalent to its contrapositive.

Contrapositive: If x²>1 then, x is not a real number such that 0<x<1.

(i) Definition of Converse: A conditional statement is not logically equivalent to its converse.

Converse: If the rectangle R is rhombus, then it is square.

(ii) Definition of Converse: A conditional statement is not logically equivalent to its converse.

Converse: If tomorrow is Tuesday, then today is Monday.

(iii) Definition of Converse: A conditional statement is not logically equivalent to its converse.

Converse: If you must visit Taj Mahal, then you go to Agra.

(iv) Definition of Converse: A conditional statement is not logically equivalent to its converse.

Converse: If the triangle is right triangle, then the sum of the squares of two sides of a triangle is equal to the square of third side.

(v) Definition of Converse: A conditional statement is not logically equivalent to its converse.

Converse: If the triangle is equilateral, then all three angles of the triangle are equal.

(vi) Definition of Converse: A conditional statement is not logically equivalent to its converse.

Converse: if 2x=3y then x:y=3:2

(vii) Definition of Converse: A conditional statement is not logically equivalent to its converse.

Converse: If the opposite angles of an quadrilateral are supplementary, then S is cyclic.

(viii) Definition of Converse: A conditional statement is not logically equivalent to its converse.

Converse: If x is neither positive nor negative then x=0

(ix) Definition of Converse: A conditional statement is not logically equivalent to its converse.

Converse: If the ratio of corresponding sides of two triangles are equal, then triangles are similar.

(i) Quantifiers means a phrase like ‘there exist’,’for all’ and ‘for every’ etc. and these are used to make the prepositional statement.

In the given statement “There exists a triangle which is not equilateral”

Quantifier is “There exist”

Hence, There exist is quantifier.

(ii) Quantifiers means a phrase like ‘there exist’,’for all’ and ‘for every’ etc. and these are used to make the prepositional statement.

In the given statement “For all real numbers x and y, xy = yx.”

Quantifier is “For all”

Hence, ‘For all’ is quantifier.

(iii) Quantifiers means a phrase like ‘there exist’,’for all’ and ‘for every’ etc. and these are used to make the prepositional statement.

In the given statement “There exists a real number which is not a rational number.”

Quantifier is “There exist”

Hence, ‘There exist’ is quantifier.

(iv) Quantifiers means a phrase like ‘there exist’,’for all’ and ‘for every’ etc. and these are used to make the prepositional statement.

In the given statement “For every natural number x, x + 1 is also a natural number.”

Quantifier is “For every”

Hence, ‘For every’ is quantifier.

(v) Quantifiers means a phrase like ‘there exist’,’for all’ and ‘for every’ etc. and these are used to make the prepositional statement.

In the given statement “For all real numbers x with x > 3, x2 is greater than 9.”

Quantifier is “For all”

Hence, ‘For all’ is quantifier.

(vi) Quantifiers means a phrase like ‘there exist’,’for all’ and ‘for every’ etc. and these are used to make the prepositional statement.

In the given statement “There exists a triangle which is not an isosceles triangle.”

Quantifier is “There exist”

Hence, ‘There exist’ is quantifier.

(vii) Quantifiers means a phrase like ‘there exist’,’for all’ and ‘for every’ etc. and these are used to make the prepositional statement.

In the given statement “For all negative integers x, x³ is also a negative integers.”

Quantifier is “For all”

Hence, ‘For all’ is quantifier.

(viii) Quantifiers means a phrase like ‘there exist’,’for all’ and ‘for every’ etc. and these are used to make the prepositional statement.

In the given statement “There exists a statement in above statements which is not true.”

Quantifier is “There exist”

Hence, ‘There exist’ is quantifier.

(ix) Quantifiers means a phrase like ‘there exist’,’for all’ and ‘for every’ etc. and these are used to make the prepositional statement.

In the given statement “There exists a even prime number other than 2.”

Quantifier is “There exist”

Hence, ‘There exist’ is quantifier.

(x) Quantifiers means a phrase like ‘there exist’,’for all’ and ‘for every’ etc. and these are used to make the prepositional statement.

In the given statement “There exists a real number x such that x² + 1 = 0.”

Quantifier is “There exist”

Hence, ‘There exist’ is quantifier.

We have given, n³-n

Let us Assume, n is even

Let n=2k, where k is natural number

n³-n=(2k)³-(2k)

n³-n=2k(4k²-1)

let k(4k²-1)=m

n³-n=2m

Therefore, (n³-n) is even.

Now, Let us Assume n is odd

Let n=(2k+1), where k is natural number

n³-n=(2k+1)³-(2k+1)

n³-n= (2k+1)[(2k+1)²-1]

n³-n= (2k+1)[(4k²+4k+1-1)]

n³-n= (2k+1)[(4k²+4k)]

n³-n= 4k(2k+1)(k+1)

n²-n= 2.2k(2k+1)(k+1)

let λ=2k(2k+1)(k+1)

n³-n=2λ

therefore, n³-n is even.

Hence, n³-n is always even

p: 125 is divisible by 5 and 7

Let,

q: 125 is divisible by 5.

r: 125 is divisible 7.

Here, q is true and r is false.

Therefore, qᴧr is False

Hence, p is not valid.

q : 131 is a multiple of 3 or 11

Let,

P: 131 is a multiple of 3.

Q: 131 is a multiple of 11.

Here, P is false and Q is False

Therefore, P ⋁ Q is False

Hence, q is not valid

Let p is false, as the sum of an irrational number and a rational number is irrational.

Let is irrational and n is rational number

But, we know that is irrational where as (r-n) is rational which is contradiction.

Here, Our Assumption is False

Hence, P is true.

We have Given for any real number x, y if x=y

To Find: x²=y²

Explanation: Let us Assume

p: x=y where x and y are real number

On squaring both sides we get

x²=y² : q (Assumption)

Therefore,

P=q

Hence, Proved

Let us Assume

p: n² is an even integer.

~p: n is not an even integer

q: n is also an even integer

~q=n is not an even integer.

Since, In the contrapositive, a conditional statement is logically equivalent to its contrapositive.

Therefore,

~q → ~p = If n is not an even integer then n² is not an even integer.

Hence, ~q is true → ~p is true.

A statement is an assertive (declarative) sentence if it is either true or false but not both.

Here, 6 is a natural number is true

To given order like Come here, Go there are not statements.

In the statement “2 + 7 > 9 or 2 + 7 < 9”

Since, Or is connecting two statement.

In the statement “Earth revolves round the Sun and Moon is a satellite of earth” And is connective.

If the statement is p then its negation is ~p, it means if p is true then ~p is false and vice versa.

Since, The negation of “A circle is an ellipse “ is “A circle is not an ellipse”

If the statement is p then its negation is ~p, it means if p is true then ~p is false and vice versa.

Since, The negation of “7 is greater than 8 “ is “7 is not greater than 8”

If the statement is p then its negation is ~p, it means if p is true then ~p is false and vice versa.

p: 72 is divisible by 2 and 3

q: 72 is divisible by 2

~q: 72 is not divisible by 2

r: 72 is divisible by 3

~r: 72 is not divisible by 3

Now,

~(qᴧr)=~q V ~r

Hence, 72 is not divisible by 2 or 72 is not divisible by 3.

If the statement is p then its negation is ~p, it means if p is true then ~p is false and vice versa.

p: Plants take in CO₂ and give out O₂

q: Plants take in CO₂

~q: Plants do not take in CO₂

r: Plants give out O₂

~r: Plants do not give out O₂

Now,

~(qᴧr)=~q V ~r

Hence, Plants do not take in CO₂ or do not give out O₂.

If the statement is p then its negation is ~p, it means if p is true then ~p is false and vice versa.

p: Rajesh or Rajni lived in Bangalore

q: Rajesh lived in Bangalore

~q: Rajesh did not lived in Bangalore

r: Rajni lived in Bangalore

~r: Rajni did not lived in Bangalore.

Now,

~(qVr)=~q ᴧ ~r

Hence, Rajesh did not live in Bangalore and Rajni did n ot live in Bangalore.

If the statement is p then its negation is ~p, it means if p is true then ~p is false and vice versa.

q: 101 is not a multiple of 3

~q: 101 is a multiple of 3

In the contrapositive, a conditional statement is logically equivalent to its contrapositive.

Since,

p: 7 is greater than 5

~p: 7 is not greater than 5

q: 8 is greater than 6

~q: 8 is not greater than 6

Therefore,

~p→ ~q = If 8 is not greater than 6, then 7 is not greater than 5.

A conditional statement is not logically equivalent to its converse.

Since,

p: x>y

q: x+a > y+a

Therefore,

p→ q

Converse of the above statement is q→ p is

Therefore, if x+a >y+a then x>y

Let p:Sun is not shining.

q:Sky is filled with clouds.

So, The converse of the statement p→ q is q→ p.

Therefore,

q→ p : If sky is filled with clouds, then the sun is not shining.

Here the statement is “If p, then q”

i.e p→ q

Contrapositive of the statement p→ q is (~q)→ (~p)

Therefore,

If ~q, then ~p

Hence, the correct option is (C)

Let p: x² is not even

q: x is not even

So, The converse of the statement p→ q is q→ p

Therefore,

If x is not even, then x² is not even.

Let p:Chandigarh is the capital of Punjab

q: Chandigarh in India.

~p: Chandigarh is not Capital of Punjab

~q: Chandigarh is not in India.

Since,

If (~q), then (~p)

Therefore,

If chandigarh is not in India, then Chandigarh is not the capital is not the capital of Punjab.

We know that p→ q is same as p only if q.

The negation of the statement is “ It is false that the product of 3 and 4 is 9”

The negation of the given statement is false.

Since, It is false that a natural number is not greater than zero.

Hence, the correct option is (C)

In the conjuction, we use “and” between two statement like p and q.

Hence, None of the given statements separated by and.

(i) It is a statement because the given statement “The angles opposite to equal sides of a triangle are equal” is true.

(ii) It is a statement because the given statement “The moon is a satellite of earth” is true.

(iii) It is not a statement because the given sentence “May God bless you!” is an exclamatory sentence.

(iv) It is a statement because the given statement “Asia is a continent” is true.

(v) It is not a statement because the given sentence “ How are you ?” is question.