Mathematical Reasoning Test

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Mathematical Reasoning Test - 9

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Question 1
1 / -0

The negation of the boolean expression
$$\sim s\vee \left( \sim r\wedge s \right) $$ is equivalent to:

A
$$r$$

B
$$s\wedge r$$

C
$$s\vee r\quad $$

D
$$\sim s\wedge \sim r\quad $$

Solution

$$\sim \left( \sim s\vee (\sim r\wedge s) \right) $$

$$s\wedge (r\vee \sim s)$$
$$(s\wedge r)\vee (s\wedge \sim s)$$

$$(s\wedge r)\vee (F)$$

$$(s\wedge r)$$
Question 2
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Negation of the statement:
$$\sqrt{5}$$ is an integer or $$5$$ is irrational is?

A
$$\sqrt{5}$$ is not an integer or $$5$$ is not irrational

B
$$\sqrt{5}$$ is not an integer and $$5$$ is not irrational

C
$$\sqrt{5}$$ is irrational or $$5$$ is an integer

D
$$\sqrt{5}$$ is an integer and $$5$$ is irrational

Solution

Let
$$p=\sqrt5$$ is an integer $$\Rightarrow$$ $$\sim p=$$ $$\sqrt5$$ is not an integer
$$q= 5 $$ is irrational $$\Rightarrow$$ $$\sim q=$$ $$5$$ is not an irrational

1) $$\sqrt{5}$$ is not an integer and $$5$$ is not an irrational number
$$\{\sim (p \vee q) = \sim p \wedge \sim q\}$$
Question 3
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Consider the following three statements:
P : 5 is a prime number.
Q : 7 is a factor of 192.
R : L.C.M. of 5 and 7 is 35.
Then the truth value of which one of thefollowing statements is true ?

A
$$(P \wedge Q)\vee (\sim R)$$

B
$$(\sim P)\wedge (\sim Q \wedge R)$$

C
$$(\sim P) \vee (Q\wedge R)$$

D
$$P\vee (\sim Q \wedge R)$$

Solution

P is True
Q is False
R is True
Option 4) $$T\vee(T\wedge T)=T$$
Question 4
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The negation of the statement: "If I become a teacher, then I will open a school" is

A
I will become a teacher and I will not open a school.

B
Either I will not become a teacher or I will not open a school.

C
Neither I will become a teacher nor I will open a school.

D
I will not become a teacher or I will open a school.

Solution

Let $$p:$$ I become a teacher

$$q:$$ I will open a school

The given statement is $$p\rightarrow q=\left( \sim p \right) \vee q$$

It negation is $$(~ \left( \text{~ p} \right) \vee q)=p\wedge \left( \text{~ q} \right) $$

Thus negation of the given statement is 'I will become a teacher and I will not open school'.
Question 5
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The contrapositive of the statement 'I go to school if it does not rain' is:

A
If it rains, I do not go to school.

B
If I do not go to school, it rains.

C
If it rains, I go to school.

D
If i go to school, it rains.

Solution

In the given statement, let $$p$$ denote the part "it does not rain"
and $$q$$ denote the part "i go to school"
So the given statement is $$p\longrightarrow q$$
Now for a contrapositive statement, by definition we have
$$\left( p\longrightarrow q \right) \leftrightarrow \left( \sim q\longrightarrow \sim p \right) $$
So $$\sim q$$ means "i do not go to school"
and $$\sim p$$ means "it rains"
$$\sim q\longrightarrow \sim p$$ means "if i do not go to school, it rains"
Question 6
1 / -0

The negation of $$\sim s \vee (\sim r\wedge s)$$ is equivalent to

A
$$S\wedge\ r$$

B
$$S\wedge \sim\ (r\wedge \sim s)$$

C
$$S\vee \sim\ (r\ \wedge \sim s)$$

D
$$None\ of\ These$$

Solution

$$\sim (\sim s \vee (\sim r\wedge s)) \Rightarrow s \: \wedge \sim (\sim r \wedge s)$$

$$\Rightarrow s \wedge (r \vee \sim s)$$

$$\Rightarrow s \wedge r$$
Question 7
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The contrapositive of the statement 'If I am not feeling well, then I will go to the doctor' is:

A
If I am feeling well, then I will not go to the doctor

B
If I will go to the doctor, then I am felling well

C
If I will not go to the doctor, then I am feeling well

D
If I will go to the doctor, then I am not feeling well.

Solution

"If i am not feeling well, then I will go to doctor."

Let $$p$$ be "I am not feeling well" and $$q$$ be "I will go to doctor".

If $$p$$, then $$q$$.

$$p\rightarrow q = \sim q\rightarrow \sim p$$

If I will not go to doctor, then I am feeling well. .
Question 8
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The contrapositive of the following statement, "If the side of a square doubles, then its area increases four times"' is:

A
If the area of a square increases four times, then its side is not doubled

B
If the area of a square increases four times, then its side is doubled

C
If the area of a square does not increase four times, then its side is not doubled

D
If the side of a square is not doubled, then its area does not increase four times

Solution

Contrapositive of $$p\rightarrow q$$ is given by $$\sim q \rightarrow \sim p$$
So, $$(3)$$ is the right option.
Question 9
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Consider the following two statements:
P: If $$7$$ is an odd number, then $$7$$ is divisible by $$2$$.
Q: If $$7$$ is a prime number, then $$7$$ is an odd number.
If $$V_{1}$$ is the truth value of the contrapositive of P and $$V_{2}$$ is the truth value of contrapositive of Q, then the ordered pair $$(V_{1}, V_{2})$$ equals:

A
$$(F, F)$$

B
$$(T, T)$$

C
$$(T, F)$$

D
$$(F, T)$$

Solution

The truth value of statement $$P$$ is $$F$$ as a True statement cannot
imply a false statement.

The truth value of statement $$Q$$ is $$T$$ as True statement implying True statement is $$T$$

The truth value of a conditional and its contra-positive are logically equivalent.
Hence $$V_1 $$ is $$F$$ and $$V_2$$ is $$T$$
Question 10
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The Boolean expression $$( ( p \wedge q ) \vee ( p \vee \sim q ) ) \wedge ( \sim p \wedge \sim q )$$ is equivalent to :

A
$${ p }\wedge (\sim { q })$$

B
$${ p }\vee (\sim { q })$$

C
$$(\sim { p })\wedge (\sim { q })$$

D
$$p\wedge q$$

Solution

By Using Truth Tables for the mentioned Boolean expression we prove that the truth table for $$(\sim { p })\wedge (\sim { q })$$ mathces.
Hence the correct answer is Option C

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Mathematical Reasoning Test - 9

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Mathematical Reasoning Test - 9

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Question 1

$$r$$

$$s\wedge r$$

$$s\vee r\quad $$

$$\sim s\wedge \sim r\quad $$

Solution

Question 2

$$\sqrt{5}$$ is not an integer or $$5$$ is not irrational

$$\sqrt{5}$$ is not an integer and $$5$$ is not irrational

$$\sqrt{5}$$ is irrational or $$5$$ is an integer

$$\sqrt{5}$$ is an integer and $$5$$ is irrational

Solution

Question 3

$$(P \wedge Q)\vee (\sim R)$$

$$(\sim P)\wedge (\sim Q \wedge R)$$

$$(\sim P) \vee (Q\wedge R)$$

$$P\vee (\sim Q \wedge R)$$

Solution

Question 4

I will become a teacher and I will not open a school.

Either I will not become a teacher or I will not open a school.

Neither I will become a teacher nor I will open a school.

I will not become a teacher or I will open a school.

Solution

Question 5

If it rains, I do not go to school.

If I do not go to school, it rains.

If it rains, I go to school.

If i go to school, it rains.

Solution

Question 6

$$S\wedge\ r$$

$$S\wedge \sim\ (r\wedge \sim s)$$

$$S\vee \sim\ (r\ \wedge \sim s)$$

$$None\ of\ These$$

Solution

Question 7

If I am feeling well, then I will not go to the doctor

If I will go to the doctor, then I am felling well

If I will not go to the doctor, then I am feeling well

If I will go to the doctor, then I am not feeling well.

Solution

Question 8

If the area of a square increases four times, then its side is not doubled

If the area of a square increases four times, then its side is doubled

If the area of a square does not increase four times, then its side is not doubled

If the side of a square is not doubled, then its area does not increase four times

Solution

Question 9

$$(F, F)$$

$$(T, T)$$

$$(T, F)$$

$$(F, T)$$

Solution

Question 10

$${ p }\wedge (\sim { q })$$

$${ p }\vee (\sim { q })$$

$$(\sim { p })\wedge (\sim { q })$$

$$p\wedge q$$

Solution

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