Mathematical Reasoning Test

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Mathematical Reasoning Test - 15

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Question 1
1 / -0

$$P\rightarrow (q\rightarrow r)$$ is logically equivalent to

A
$$(q\vee q)\rightarrow \sim r$$

B
$$(p\wedge q)\rightarrow \sim r$$

C
$$(p\vee q)\rightarrow r$$

D
$$(p\wedge q)\rightarrow r$$

Solution

$$ P \rightarrow (\sim q\rightarrow r)$$
$$ = p \rightarrow (\sim q\vee r)$$
$$ \sim p\vee (\sim q\vee r) $$
$$ (\sim p \vee \sim q) \vee r$$
$$ \sim (p\wedge q) \vee r$$
$$ (p\wedge q) \rightarrow r$$
Question 2
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$$p\vee q$$ is true, when

A
either $$p$$ of $$q$$ are true

B
$$p$$ is true and $$q$$ is false

C
$$p$$ is false and $$q$$ is true

D
All of these

Solution

We draw the table q disjunctive between two variables

$$p\quad \quad q\quad \quad pVq\\ T\quad \quad T\quad \quad T\\ T\quad \quad F\quad \quad T\\ F\quad \quad T\quad \quad T\\ F\quad \quad F\quad \quad F$$

The disjunction is true when either $$p$$ is true, $$q$$ is true or both $$p$$ and $$q$$ is true.
Question 3
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Contrapositive of the statement ''if two number are not equal then their square are not equal is ;

A
If the squares of two number are equal , then the number are not equal

B
If the squares of two number are equal , then the number are equal

C
If the square of two number are not equal then number are equal

D
If the square of two number are not, equal , then the number are not equal

Solution

Contrapositive of $$p \rightarrow q$$ is $$\sim q \rightarrow \sim p$$
Hence, contrapositive of $$\text{if two number are not equal then their square are not equal is}$$
$$\text{If the squares of two number are equal, then the number are equal}$$.
Question 4
1 / -0

Which of the following is not a tautology?

A
$$p\rightarrow (p\vee q)$$

B
$$(p\wedge q)\rightarrow p$$

C
$$(p\vee q)\rightarrow (p\wedge (\sim q))$$

D
$$(p\vee \sim p)$$

Solution

$$(1)$$ $$p\rightarrow (p\vee q)$$
$$\sim p\vee (p\vee q)=t$$
$$(2)$$ $$\sim(p\wedge q)\vee P$$
$$(\sim p\vee \sim q)\vee p=t$$
$$(3)$$ $$\sim(p\vee q)\vee (p\wedge \sim q)$$
$$=(\sim p\wedge \sim q)\vee (p\wedge \sim q)$$
$$=(\sim p\vee p)\wedge \sim q$$
$$=t\wedge \sim q=\sim q\neq t$$
$$(4)$$
Alter:
(a)
p q $$p\vee q$$ $$p\rightarrow p\vee q$$
T T T T
T F T T
F T T T
F F F T
(b)
p q $$p\vee q$$ $$p\rightarrow p\vee q$$
T T T T
T F F T
F T F T
F F F T
(c)
p q $$p\vee q$$ $$\sim q$$ $$p\wedge \sim q$$ $$p\vee q\rightarrow p\wedge (\sim q)$$
T T T F F F
T F F T T T
F T F F F T
F F F T F T
Question 5
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Select and write the correct answer from the given alternative of the following question:
If $$ p \wedge q $$ is false and $$ p \wedge q $$ is true, then __________ is not true.

A
$$ p \vee q $$

B
$$ p \leftrightarrow q $$

C
$$ \sim p\, \vee \, \sim q $$

D
$$ q \vee \, \sim p $$

Solution

If $$ p \wedge q $$ is false and $$ p \vee q $$ is true, then $$ p \leftrightarrow q $$ is not true.
Question 6
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Select and write the correct answer from the given alternative of the following question:
Inverse of statement pattern $$ (p \vee q) \rightarrow (p \wedge q) $$ is ________.

A
$$ (p \wedge q) \rightarrow (p \vee q) $$

B
$$ \sim (p \vee q) \rightarrow (p \wedge q) $$

C
$$ (\sim p \wedge \, \sim q) \rightarrow (\sim p \vee \, \sim q) $$

D
$$ (\sim p \vee \, \sim q) \rightarrow (\sim p \, \wedge \, \sim q) $$

Solution

Inverse of statement pattern $$ (p \vee q) \rightarrow (p \wedge q) $$ is $$ (\sim p \wedge \, \sim q) \rightarrow (\sim p \vee \, \sim q) $$
Question 7
1 / -0

Select and write the correct answer from the given alternative of the following question:
The negation of inverse of $$ \sim p \rightarrow q $$ is ___________.

A
$$ q \wedge p $$

B
$$ \sim p \wedge\, \sim q $$

C
$$ p \wedge q $$

D
$$ \sim q \rightarrow \, \sim p $$

Solution

The negation of inverse of $$ \sim p \rightarrow q $$ is $$ q \wedge p $$
Question 8
1 / -0

Select and write the correct answer from the given alternative of the following question:
If $$ p \wedge q $$ is F , $$ p \rightarrow q $$ is F then the truth values of $$p$$ and $$q$$ are ___________.

A
T , T

B
T , F

C
F , T

D
F , F

Solution

If $$ p \wedge q $$ is F , $$ p \rightarrow q $$ is F then the truth values of $$p$$ and $$q$$ are T , F.
Question 9
1 / -0

Select and write the correct answer from the given alternative of the following question:
If $$ A = \left \{1 , 2 , 3 , 4 , 5 \right \} $$ then which of the following is not true ?

A
$$ \exists \, x \, \in \, A $$ such that $$ x + 3 = 8 $$

B
$$ \exists \, x \, \in \, A $$ such that $$ x + 2 < 9 $$

C
$$ \forall \, x \, \in \, A , x + 6 \geq 9 $$

D
$$ \exists \, x \, \in \, A $$ such that $$ x + 6 < 10 $$

Solution

$$ \forall \, x \, \in \, A , x + 6 \geq 9 $$ as,
$$ \because 1+6 \ngeq 9$$ , $$ 2+6 \ngeq 9$$
Question 10
1 / -0

Select and write the correct answer from the given alternative of the following question:
The negation of $$ p \wedge (q \rightarrow r) $$ is __________.

A
$$ \sim p \wedge (\sim q \rightarrow \, \sim r) $$

B
$$ p \vee (\sim q \vee r) $$

C
$$ \sim p \wedge (\sim q \rightarrow \, \sim r) $$

D
$$ \sim p \vee (\sim q \wedge \sim r) $$

Solution

The negation of $$ p \wedge (q \rightarrow r) $$ is $$ \sim p \vee (\sim q \wedge \sim r) $$

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Re-Attempt Weekly Quiz Competition

p	q	$$p\vee q$$	$$\sim q$$	$$p\wedge \sim q$$	$$p\vee q\rightarrow p\wedge (\sim q)$$
T	T	T	F	F	F
T	F	F	T	T	T
F	T	F	F	F	T
F	F	F	T	F	T

Mathematical Reasoning Test - 15

Result

Mathematical Reasoning Test - 15

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SHARING IS CARING

Question 1

$$(q\vee q)\rightarrow \sim r$$

$$(p\wedge q)\rightarrow \sim r$$

$$(p\vee q)\rightarrow r$$

$$(p\wedge q)\rightarrow r$$

Solution

Question 2

either $$p$$ of $$q$$ are true

$$p$$ is true and $$q$$ is false

$$p$$ is false and $$q$$ is true

All of these

Solution

Question 3

If the squares of two number are equal , then the number are not equal

If the squares of two number are equal , then the number are equal

If the square of two number are not equal then number are equal

If the square of two number are not, equal , then the number are not equal

Solution

Question 4

$$p\rightarrow (p\vee q)$$

$$(p\wedge q)\rightarrow p$$

$$(p\vee q)\rightarrow (p\wedge (\sim q))$$

$$(p\vee \sim p)$$

Solution

Question 5

$$ p \vee q $$

$$ p \leftrightarrow q $$

$$ \sim p\, \vee \, \sim q $$

$$ q \vee \, \sim p $$

Solution

Question 6

$$ (p \wedge q) \rightarrow (p \vee q) $$

$$ \sim (p \vee q) \rightarrow (p \wedge q) $$

$$ (\sim p \wedge \, \sim q) \rightarrow (\sim p \vee \, \sim q) $$

$$ (\sim p \vee \, \sim q) \rightarrow (\sim p \, \wedge \, \sim q) $$

Solution

Question 7

$$ q \wedge p $$

$$ \sim p \wedge\, \sim q $$

$$ p \wedge q $$

$$ \sim q \rightarrow \, \sim p $$

Solution

Question 8

T , T

T , F

F , T

F , F

Solution

Question 9

$$ \exists \, x \, \in \, A $$ such that $$ x + 3 = 8 $$

$$ \exists \, x \, \in \, A $$ such that $$ x + 2 < 9 $$

$$ \forall \, x \, \in \, A , x + 6 \geq 9 $$

$$ \exists \, x \, \in \, A $$ such that $$ x + 6 < 10 $$

Solution

Question 10

$$ \sim p \wedge (\sim q \rightarrow \, \sim r) $$

$$ p \vee (\sim q \vee r) $$

$$ \sim p \wedge (\sim q \rightarrow \, \sim r) $$

$$ \sim p \vee (\sim q \wedge \sim r) $$

Solution

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