MATHS MCQ

1. T, T, F

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

If you drink your milk is negation of the hypothesis if you donot drink your milk you are strong is negation of the conclusion you will not be strong.

The converse of "If P then Q" is "If Q then P".

8 is less than 6 if false. So it is a statement.
Every set is finite set is false. So it is a statement.
The sun is a star is true. So it is a statement.
Mathematics is fun. This sentence is not always true. Hence, it is not a statement

Contrapositive statement for given statement “if p then q” is “if not q then not p” i.e. ~q ⇒ ~p.
So, contrapositive for “If x is divisible by 4 then x must be divisible by 2” is “If x is not divisible by 2 then x is not divisible by 4”.

1. a tautology

(p or q) is false when both p and q are false otherwise it is true.

The statement 3 is a prime number is true.
So, it is a statement.

2. false

“Close the door” is not a statement as it is an imperative sentence.
“11 comes after 12” is a statement as it is true.
“India is a beautiful country” is not a statement as opinion of beautifulness vary from person to person.
“This is useless” is not a statement as it involves this which doesn’t give any idea about what we are talking.

Let p: 101 is not multiple of 3.
~p: 101 is a multiple of 3.

4. ∼r → (∼p ∧ ∼q)

(p and q) is true when both p and q are true otherwise it is false.

2. (p ∧ ∼q) ∧ ∼p

Converse statement for given statement “if p then q” is “if q then p” i.e. q ⇒ p or q implies p.

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

Since, statement is a sentence which is ether true or false. So, 6 is a natural number which is true.

We have, p: 72 is divisible by 2 and 3.
Let q: 72 is divisible by 2.
r: 72 is divisible by 3.
~q: 72 is not divisible by 2.
~r. 72 is not divisible by 3.
~{q ⋀ f ) -~q v ~r
⟹ 72 is not divisible by 2 or 72 is not divisible by 3.

2. T, F, F

Given, p → q
Now, conditional of the statement is
p only if q

2. ∼p ∧ q

2. ∼p ∧ ∼q

A compound statement is made up of two or more statements where each statement is known as component statement.

A compound statement is connected with And, or, etc.
So, the statement 7 is both odd and prime number is a compound statement.

p : A number is prime
q : It is odd
We have p → q
The inverse of p → q is ∼ p → ∼ q
i.e. If a number is not a prime then it is not odd.

Both Ram and Shyam are tall.

Contrapositive statement for given statement “if p then q” is “if not q then not p" i.e. ~q ⇒ ~p.

If p then q means p only if q or p ⇒ q which is not same as q only if p or q ⇒ p.

If p then q means p ⇒ q or negation of q implies negation of p i.e. ~q ⇒ ~p.

Of the given options only 4 < 5, i.e; option A satisfies x < 5.

The sentences in (b), (c), and (d) are neither true nor false.All these sentences are pieces of advice.
Sentence (a) is a definite statement.

If its raining then there must be cloud in the sky.
This a statement.
A statement is a closed sentence. It can also be a mathematical identity.
A statement should have a complete meaning independently.

The negation of the given statement is "he is not rich or not happy".

Statements generally not use ambiguous words like here, there, this, that, today, tonight, tomorrow.

4. T, T, F

Let p : I become a teacher
q : I will open a school
The given statement is p → q = (∼p) ∨ q
It negation is (p) ∨ q) = p ∧ (q)
Thus negation of the given statement is 'I will become a teacher and I will not open school'.

Contrapositive statement for given statement “if p then q” is “if not q then not p” i.e. ~q ⇒ ~p.
So, contrapositive for “If x is divisible by 4 then x must be divisible by 2” is “If x is not divisible by 2 then x is not divisible by 4”.

The denial of statement is known as negation of the statement.
It is denoted by ~p if statement is denoted by p.

We know, If a statement is true then its truth value is T and if statement is false then F.

1. T, F, F, F

In direct method, we assume p to be true and use it to prove that q is also true.
There is no case for p to be false in direct method.

Converse statement for given statement “if p then q” is “if q then p” i.e. q ⇒ p.
So, converse for “If x is divisible by 4 then x must be divisible by 2” is “If x is divisible by 2 then x is must be divisible by 4”.

3. T, T

Let p: x > y
q: x + a > y + a
P ⟶ q
Converse of the above statement is:
q ⟶ P
i.e., If x + a > y + a, then x > y

Converse statement for given statement “if p then q” is “if q then p” i.e. q ⇒ p or q implies p.

The methods that are used to check the validity of statements include the following.

1. direct method
2. contrapositive method
3. method of contradiction
4. using a counter example

3. (p Λ q) Λ (p Λ ~r)

If p then q means p is sufficient condition for q or p ⇒ q.
It is not same as q is sufficient condition for p.

The statement 6 is a natural number is true.
So, it is a statement.

2. (p → q)

4. A quadrilateral is a square and it is not a rhombus.

1. T, F, F

Given, statement is The product of 3 and 4 is 9
The negation of the statement is:
It is false that the product of 3 and 4 is 9

Let p: Chandigarh is capital of Punjab.
and q: Chandigarh is in India.
~p: Chandigarh is not capital of Punjab.
~q: Chandigarh is not in India.
Contra positive of the statement p → q
if (~q), then (~p).
It Chandigarh is not in India, then Chandigarh is not the capital of Punjab.

“Product of 1 and 2 is -4” is a statement as it is false. We know, product of 1 and 2 is 2.
“Squares are always positive” is a statement as it is true.
“Sum of 2 and 3 is 9” is a statement as it is false.
But “Give me a cup of tea” is not a statement as it is an order so it is imperative sentence.

1. a contradiction

Converse statement for given statement “if p then q” is “if q then p” i.e. q ⇒ p.
So, converse for “If x is divisible by 4 then x must be divisible by 2” is “If x is divisible by 2 then x is must be divisible by 4”.

Sentence involving variable time such as today, tomorrow, or yesterday are not statements. This is because it is not known what time is referred here.

3. If x - 2y = 9, then x ≠ 5 or y ≠ - 2

Given Statement:
The solution is easy if you read the question carefully.
Logically equivalent statement:
If you do not read the question carefully, the solution is hard.

“0 is a positive or negative number” is formed by connecting the given statements using or.

Let p: A circle is an ellipse.
~p: A circle is not an ellipse.

Negation of "Every Natural number is an integer". is "Every Natural number is NOT an integer".

When we talk about this, that, here, there, we are not sure about what we are talking about so “There are 27 days in this month” is not a statement.
“February has 28 days” and “February has 29 days” both are not statements because February may have 28 days or 29 das based on the year. “There are 32 days in this month” is a statement as it is false.
We cannot have 32 days in a month.

p → q
If p, then q
Contra positive of the statement p → q is (~q) →(~p).
If ~q, then ~p.

Connective word is “and”.

Given statement is:
A natural number is greater than zero

2. r → (p ∨ ∼q)

1. True

4. Conditional, Contrapositive

If p then q means p is sufficient condition for q or p ⇒ q.
It is not same as q is sufficient condition for p.

1. I play cricket during the holidays and just sleep throughout the day.

“Two and two makes four” and “Elephant is heavier than ant” are true so they are statements. “A prime number is always odd” is false as prime number may be even so it is a statement. “Sum of and b is 5” is not a statement as it can be true or false based on the values of a and b taken.

3. (~p Λ ~q) ∨ (~q Λ r)

Sometimes in mathematics, it's important to determine what the opposite of a given mathematical statement is. This is usually referred to as "negating" a statement.
One thing to keep in mind is that if a statement is true, then its negation is false (and if a statement is false, then its negation is true).
So, If (something is done), then (something happens),
Negation: If (something is done), then (something does not happen),
you believe in yourself and are honest and did not get success.

3. T, F, F, T

A sentence is called mathematically acceptable statement if it is either true or false but not both.

In ‘2 + 7 > 9 or 2 + 7 < 9’ the connective is ‘or’.

If p then q means p implies q i.e. p ⇒ q.
This does not mean q ⇒ p.
Q only if p means q ⇒ p.

Let p, q, r be three statements defined as
p : a number N is divisible by 15
q : number N is divisible by 5
r : number N is divisible by 3
Here given statement is p → (q∨r)
Here negative of above statement is
∼ (p → (q∨r)) ≡ p ∧ ( ∼ (q∨r)
≡ p ∧ (∼ q∧∼ r)
i.e. A number is divisible by 15 and it is not divisible by 5 and 3.

A sentence is a statement if it is true.
None of the above sentence is true.

Deb and Sam go to the mall only if it is snowing which means if it is not snowing, they don't go to the mall.

2. x = 4, 2x + 3 ≠ 9

As we know, the converse of a statement p ⇒ q is the statement q ⇒ p.
So, the converse of the statement “If x > y, then x + a > y + a” is.
If x + a > y + a, then x > y.

The negation of the given statement is false. i.e., It is false that a natural number is not greater than zero.

Take p : x has courage
and q : x will win
So the given conjugation is p ⇒ q
Now we need to find converse of this.
Be definition, Converse will be q ⇒ p
This is symbolic for "If x wins then x has courage

Given,
Statement: Akash or Ankitha lived in Goa
Negation of the above statement is:
Akash did not live in Goa and Ankitha did not live in Goa.

Smoking is injurious to health.
It is a statement.
2 + 2 = 4; It is a mathematical statement.
2 is the only even prime number.
Mathematical statement.
Come here.
It is not a statement but it is an order.

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

3. Contradiction

1. TFFF

For a statement with ‘and’, if both statements are true then the given statement will be true.
In statement “25 is divisible by 2 and 3”, 25 is divisible by 2 is true but 25 is divisible by 3 is false so it is false statement.
In statement “12 is a positive and prime number”, 12 is a positive number is true but 12 is a prime number is false so it is a false statement.
In statement “13 is a positive and odd number”, both statements i.e. 13 is a positive number and 13 is an odd number are true so it is a true statement.
In statement “49 is a square and a cube”, 49 is a square is true but 49 is a cube is false so it is a false statement.

A person is not mean is the negation of the hypothesis that a person is mean and they are not a fighter is the negation of the conclusion they are a fighter.

2. If a number is not a prime then it is odd

Prime number is not divisible by 2 always.
Two and two makes our not five.
All integers are not positive.
100 is divisible by 2 and 5 both. So, component statements of “100 is divisible by 2 and 5” are true.

The statement is true if Mary is thinking of:

1. As, 6 is divisible by both 2 and 3.
2. As, 8 is divisible by 2.
3. As, 15 is divisible by 3.

Hence, the statement is false for option (C) as 11 is not divisble by either 2 or 3.

Contrapositive statement for given statement “if p then q” is “if not q then not p” i.e. ~q ⇒ ~p.

2. true

If p then q statement will be true.
If p is true then q must be true and if q is true then p must be true.

To find the inverse we need to negate the hypothesis and conclusion. On negating the hypothesis we get you reason deductively and on negating the conclusion we get you understand geometry.

A sentence is called mathematically acceptable statement if it is either true or false but not both
“Two plus ''two is three” is false so it is a statement.
Rest we cannot decide whether they are true or false.

In contrapositive method, we assume p to be false and use it to prove that q is also false.
There is no case for p to be true in direct method.

If p then q means p ⇒ q or negation of q implies negation of p i.e. ~q ⇒ ~p.

To given order cannot be a statement. So ‘Come here’ is not a statement.

If p then q means q is necessary condition for p or p ⇒ q.
It is not same as p is necessary condition for q.

4. s ∧ r

Take p : x has courage
And q : x will win
So the given conjugation is p ⇒ q
Now we need to find Inverse of this.
Be definition, Inverse of p ⇒ q is ∼ p ⟶ ∼ q
∼ p : x has no courage,
∼ q : x will not win
Thus the inverse ∼ p ⟶ ∼ q is symbolic form of "If x has no courage, then x will not win.

The statement is true as it's already given that All scientists working in America are talented.

“Close the door” is not a statement as it is an imperative sentence.
“11 comes after 12” is a statement as it is true.
“India is a beautiful country” is not a statement as opinion of beautifulness vary from person to person.
“This is useless” is not a statement as it involves this which doesn’t give any idea about what we are talking.

1. ~(Q → (P Λ ~R))

Let p: 7 is greater than 8.
~p: 7 is not greater than 8.

Statement: 7 is greater than 8
Negation: 7 is not greater than 8

The converse of the statement p ⇒ q is
q ⇒ p

Given, statement is:
If a triangle is not equilateral, it is not isosceles.
Now, contra-positive is:
If a triangle is equilateral, it is isosceles.

The statement Everyone in India speaks Hindi is not true.
This is because, there are some states like Tamil nadu, Kerala, etc. where the person does not speak Hindi.

Negation of statement "P" is "not P"

3. (∧, ∨)

a number is not divisible by 4 is the negation of the hypothesis a number is divisible by 4, and it is not divisible by 2 is the negation of the conclusion it is also divisible by 2.

 
1.  

Negation of greater is ‘not greater’. ‘Not greater’ means either smaller than or equal to. So, “Sum of 2 and 3 is smaller than or equal to 4”, “Sum of 2 and 3 is not greater than 4”, “Sum of 2 and 3 is not greater than 4” are negation of the given statement. And “Sum of 2 and 3 is smaller than 4” is not the negation of given statement as it should include equal to also.

2. it’s either true or false but not both.

2. (p ⇒ q) ∨ (p ⇒ r)

4. TTTT

2. Delhi is on the Jupiter

Given statement: Earth revolves around the Sun and Moon is a satellite of earth
Here, the connective word is “and.”

1. Pie zie

Solution:
It is clear that pie means light. So there can be only 2 correct options, either pie zie or pie mie.
But mie means blue as it is common in the first two statements.

Since Delhi and Mumbai both are cities in India so “Delhi and Mumbai are cities of India” is true.
Since Chandigarh is the capital of both states Punjab and Haryana so “Chandigarh is capital of Punjab and Haryana” is true.
Since 81 is a square and not a cube so “81 is a square and cube” is false.
Since 16 is even number and is also divisible by 8 so “16 is even number and divisible by 8” is true.

2. ∼q ∧ (p ∧ r)

All the statements are conjunction.

It is a fundamental concept that, denial of a mathematical statement is called its negation.

A rectangle has four sides so it is a quadrilateral. All the four sides are not equal as two are equal and other two are equal.

2. ~ (p ⇒ ~q) = ~p ∧ q

If two simple statements p and q are connected by the word ‘and’, then the resulting compound statement p and q is called a conjunction ofp and q. Here, none of the given statement is conjunction.

“Delhi is not in India”, “It is false that Delhi is in India”, “It is not the case that Delhi is in India” are negations of the given statement.
Negation of a statement can be formed by adding ‘it is false that’ or ‘it is not the case that’.
In statement “It is false that Delhi is not in India” two negation phrases are used which is wrong so it is not the negation statement.

4. a statement

4. (∼p ∨ ∼q) ∨ (p ∨ ∼q)

2. (p → q)

1. (p ∧ (p → q)) → q

4. x + 3 = 10, x ÎR.

A statement is logically equivalent to its contrapositive To form the contrapositive switch the "If" and "then" sections of the statement AND insert "NOTs" into each section Notice in this situation that inserting "NOTs" turns the thoughts positive (you are negating negative thoughts).

 
1.  

4. The statement p is true and the statement q is true.

“Today is Monday”, “Tomorrow will be holiday”, “There will be full moon tonight” are not the statements because we are not sure which day or night we are talking about.
“If today is Tuesday then tomorrow will be Sunday” is a statement because we are sure that it is false.
Wednesday come after Tuesday so if today is Tuesday then tomorrow will be Wednesday.

The given statement is True, as 4 is the first composite number. Also 2 + 5 = 7

3. [A ∧ (A → B)] → B

24 is even number so “24 is even number or odd number” is correct. 15 is divisible by 3 so “15 is prime or divisible by 3” is true.
71 is prime as well as odd so “71 is odd number or prime” is true.
Neither Japan nor China is in India so “Japan or China is in India” is false.

The contrapositive of the statement If p, then q, is If ~ q, then ~ p.

1. ∼q → ∼p

Given, statement is if a number is divisible by 10, then it is divisible by 5
Now, converse of the statement is:
If a number is divisible by 5, then it is divisible by 10

We have, p: Rajesh or Rajni lived in Bangalore.
and q: Rajesh lived in Bangalore.
r: Rajni lived in Bangalore.
~q: Rajesh did not live in Bangalore.
~r. Rajni did not live in Bangalore.
~ (q v r): Rajesh did not live in Bangalore and Rajni did not live in Bangalore.

Matt can play Hockey on any day other than Sunday.

1. T, F, F

4. x + 3 = 10, x Î R.

Given, statement is 2 + 7 > 9 or 2 + 7 < 9 Here, connective is or. It connects two statement 2 + 7 > 9, 2 + 7 < 9.

In the given statement, let p denote the part "it does not rain"
Aand q denote the part "i go to school"
So the given statement is p ⟶ q
Now for a contrapositive statement, by definition we have
(p ⟶ q) ↔ (∼ q ⟶ ∼ p)
So ∼ q means "i do not go to school"
and ∼ p means "it rains"
∼ q ⟶ ∼ p means "if i do not go to school, it rains".

p → q
Inverse is ∼p → ∼q
⇒ Inverse of:
It it is raining then the grass is wet is:
If it is not raining then the grass is not wet.

2. (p ∧ q) ∧ r

2. ∼p ∧ ∼q

4. T, F

Given statement is if p then q
Now, contra-positive of the statement is:
If ~q then ~p

Therefore, the reason comes after the assertion.
Then, the inverse of the expression "if it rains, then I do not go fishing " is:
If I go fishing, then it does not rain.

If p then q means if p is true then q must be true. It says nothing when p is false.
If p is false then q might be true or false.

4. T, T

If p then q means if p is true then q must be true.
It says nothing when p is false.
If p is false then q might be true or false.

2. If a number is not a prime then it is odd

If p then q means p implies q i.e. p ⇒ q.
This does not mean q ⇒ p.
Q only if p means q ⇒ p.

Given statement is:
A natural number is greater than zero Negation of the statement:
A natural number is not greater than zero
It is false that a natural number is greater than zero

The statement 6 is a natural number is true.
So, it is a statement.

Let p: Sun is not shining.
q: Sky is filled with clouds.
So, the converse of the statement p → q is q → p.
i.e., If Sky is filled with clouds, then the Sun is not shining.

Negation of all angles is ‘not all angles’. So, negation of the given statement is “There is at least one angle different in equilateral triangle”, “It is false that all angles are equal in equilateral triangle”, “It is not the case that all angles are equal in equilateral triangle”. But “All angles are unequal in equilateral triangle” cannot be the negation of given statement as this does not include isosceles triangles.