Mathematical Reasoning Test

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Mathematical Reasoning Test - 11

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Question 1
1 / -0

The converse of "if $$x\in A\cap B$$ then $$x\in A$$ and $$x\in B$$", is

A
If $$x\in A$$ and $$x\in B$$, then $$x\in A\cap B$$.

B
If $$x\not\in A\cap B$$, then $$x\not\in A$$ or $$x\not\in B$$.

C
If $$x\not\in A$$ or $$x\not\in B$$, then $$x\not\in A\cap B$$.

D
If $$x\not\in A$$ or $$x\not\in B$$, then $$x\in A\cap B$$.

Solution

The converse of "If P then Q" is "If Q then P"
Hence, Option A
Question 2
1 / -0

$$(p\wedge\sim p)\wedge(p\vee q)$$ is a

A
contradiction

B
tautology

C
negation

D
none

Solution

Truth table of $$(p\wedge\sim p)\wedge(p\vee q)$$:

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

we observe that $$(p\wedge\sim p)\wedge(p\vee q)$$ is always false. Hence, $$(p\wedge\sim p)\wedge(p\vee q)$$ is a contradiction.
Question 3
1 / -0

$$(\sim p\wedge q)\wedge q$$ is

A
a tautology

B
a contradiction

C
neither a tautology nor a contradiction

D
none of these

Solution

Truth table of $$(\sim p\wedge q)\wedge q$$:

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

$$\therefore(\sim p\wedge q)\wedge q$$ is neither true always nor false always.
Hence, $$(\sim p\wedge q)\wedge q$$ is neither a tautology nor a contradiction.
Question 4
1 / -0

$$\sim(p\wedge q)\equiv$$

A
$$\sim p\vee\sim q$$

B
$$ p\vee\sim q$$

C
$$\sim p\vee q$$

D
None

Solution

Truth table:

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

The truth values of $$\sim(p\wedge q)$$ and $$\sim p\wedge\sim q$$ are same.

Hence, $$\sim(p\wedge q)\equiv\sim p\vee\sim q$$
Question 5
1 / -0

Mary says "The number I am thinking is divisible by 2 or it is divisible by 3". This statement is false if the number Mary is thinking of is

A
6

B
8

C
11

D
15

Solution

The statement is true if Mary is thinking of
($$A$$) As, $$6$$ is divisible by both 2 and 3.
($$B$$) As, $$8$$ is divisible by 2.
($$D$$) As, $$15$$ is divisible by 3.
Hence, the statement is $$false$$ for option ($$C$$) as 11 is not divisble by either 2 or 3.
Question 6
1 / -0

What is true about the statement "If two angles are right angles the angles have equal measure" and its converse "If two angles have equal measure then the two angles are right angles"?

A
The statement is true but its converse is false

B
The statement is false but its converse is true

C
Both the statement and its converse are false

D
Both the statement and its converse are true

Solution

Two right angles are always equal, each measuring 90 degrees.
However, two equal angles can be anything not necessarily equal to 90 degrees always.
Hence $$A$$ is correct.
Question 7
1 / -0

If statement $$p \rightarrow (q \vee r)$$ is true then the truth values of statements p, q, r respectively

A
T, F, T

B
F, T, F

C
F, F, F

D
all of these

Solution

$$\because p \rightarrow (q \vee r)$$ is false
$$\Rightarrow$$ p is true and (q $$\vee$$ r) is false
$$\Rightarrow$$ p is true, q and r both are false
i.e. p $$\rightarrow$$ (q $$\vee$$ r) is false when truth values of p, q, r are T, F, F respectively otherwise it is true.
Question 8
1 / -0

Consider the sentence: x<5
Which of the following integers makes this open sentence true?

A
4

B
5

C
6

D
none of the above

Solution

Of the given options only $$4<5$$ ,i.e; option $$A$$ satisfies $$x<5$$
Question 9
1 / -0

Which of the following statements is the inverse of "If you do not understand geometry, then you do not know how to reason deductively."?

A
If you reason deductively, then you understand geometry.

B
If you understand geometry, then you reason deductively.

C
If the do not reason deductively, then you understand geometry.

D
None of these

Solution

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.
Question 10
1 / -0

Which of the following statements is logically equivalent to "If you live in a mansion, then you have a big heating bill."?

A
If you have a big heating bill, then you live in a mansion.

B
If you do not live in a mansion, then you do not have a big heating bill.

C
If you do not have a big heating bill, then you do not live in a mansion.

D
None of these

Solution

If you live in a mansion, then you have a big heating bill. This means that if you do not have a big heating bill, then u don't live in a mansion.
So, option $$C$$ is correct.

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

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

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

Mathematical Reasoning Test - 11

Result

Mathematical Reasoning Test - 11

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Question 1

If $$x\in A$$ and $$x\in B$$, then $$x\in A\cap B$$.

If $$x\not\in A\cap B$$, then $$x\not\in A$$ or $$x\not\in B$$.

If $$x\not\in A$$ or $$x\not\in B$$, then $$x\not\in A\cap B$$.

If $$x\not\in A$$ or $$x\not\in B$$, then $$x\in A\cap B$$.

Solution

Question 2

contradiction

tautology

negation

none

Solution

Question 3

a tautology

a contradiction

neither a tautology nor a contradiction

none of these

Solution

Question 4

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

$$ p\vee\sim q$$

$$\sim p\vee q$$

None

Solution

Question 5

6

8

11

15

Solution

Question 6

The statement is true but its converse is false

The statement is false but its converse is true

Both the statement and its converse are false

Both the statement and its converse are true

Solution

Question 7

T, F, T

F, T, F

F, F, F

all of these

Solution

Question 8

4

5

6

none of the above

Solution

Question 9

If you reason deductively, then you understand geometry.

If you understand geometry, then you reason deductively.

If the do not reason deductively, then you understand geometry.

None of these

Solution

Question 10

If you have a big heating bill, then you live in a mansion.

If you do not live in a mansion, then you do not have a big heating bill.

If you do not have a big heating bill, then you do not live in a mansion.

None of these

Solution

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