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CBSE Questions for Class 11 Engineering Maths Mathematical Reasoning Quiz 1 - MCQExams.com
CBSE
Class 11 Engineering Maths
Mathematical Reasoning
Quiz 1
The contrapositive of the following statement, "If the side of a square doubles, then its area increases four times"' is:
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If the area of a square increases four times, then its side is not doubled
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If the area of a square increases four times, then its side is doubled
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If the area of a square does not increase four times, then its side is not doubled
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If the side of a square is not doubled, then its area does not increase four times
Explanation
Contrapositive of $$p\rightarrow q$$ is given by $$\sim q \rightarrow \sim p$$
So, $$(3)$$ is the right option.
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:
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$$(F, F)$$
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$$(T, T)$$
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$$(T, F)$$
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$$(F, T)$$
Explanation
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 $$T$$
The Boolean expression $$( ( p \wedge q ) \vee ( p \vee \sim q ) ) \wedge ( \sim p \wedge \sim q )$$ is equivalent to :
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$${ p }\wedge (\sim { q })$$
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$${ p }\vee (\sim { q })$$
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$$(\sim { p })\wedge (\sim { q })$$
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$$p\wedge q$$
Explanation
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
The negation of $$\sim s \vee (\sim r\wedge s)$$ is equivalent to
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$$S\wedge\ r$$
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$$S\wedge \sim\ (r\wedge \sim s)$$
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$$S\vee \sim\ (r\ \wedge \sim s)$$
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$$None\ of\ These$$
Explanation
$$\sim (\sim s \vee (\sim r\wedge s)) \Rightarrow s \: \wedge \sim (\sim r \wedge s)$$
$$\Rightarrow s \wedge (r \vee \sim s)$$
$$\Rightarrow s \wedge r$$
The contrapositive of the statement 'If I am not feeling well, then I will go to the doctor' is:
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If I am feeling well, then I will not go to the doctor
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If I will go to the doctor, then I am felling well
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If I will not go to the doctor, then I am feeling well
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If I will go to the doctor, then I am not feeling well.
Explanation
"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. .
The contrapositive of the statement 'I go to school if it does not rain' is:
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If it rains, I do not go to school.
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If I do not go to school, it rains.
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If it rains, I go to school.
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If i go to school, it rains.
Explanation
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"
The contrapositive of the statement "If it is raining, then I will not come", is :
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If I will come, then it is not raining.
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If I will not come, then it is raining.
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If I will come, then it is raining.
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If I will not come, then it is not raining.
Explanation
Let $$p:$$It is raining
and $$q:$$ I will not come.
Given statement is $$p \rightarrow q$$
Contrapositive of $$p \rightarrow q$$ is $$\sim q \rightarrow \sim p$$
So, $$\sim p$$ : It is not raining.
$$\sim q$$: I will come.
So, the contrapositive statement is "If I will come, then it is not raining."
The contrapositive of "If $$x\in A\cup B$$, then $$x\in A$$ or $$x\in B$$", is
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If $$x\in A$$ or $$x\in B$$, then $$x\in A\cup B$$.
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If $$x\not\in A\cup B$$, then $$x\not\in A$$ and $$x\not\in B$$.
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If $$x\not\in A$$ and $$x\not\in B$$, then $$x\not\in A\cup B$$.
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If $$x\not\in A$$ and $$x\not\in B$$, then $$x\in A\cup B$$.
Explanation
Contrapositive of statement "If P then Q" is "If not Q then not P"
Hence, C.
Earth is a planet. Choose the option that is a negation of this statement.
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Earth is round
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Earth is not round
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Earth revolves round the sun
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Earth is not a planet
Explanation
Negation of statement "P" is "not P"
The converse of "if $$x\in A\cap B$$ then $$x\in A$$ and $$x\in B$$", is
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If $$x\in A$$ and $$x\in B$$, then $$x\in A\cap B$$.
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If $$x\not\in A\cap B$$, then $$x\not\in A$$ or $$x\not\in B$$.
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If $$x\not\in A$$ or $$x\not\in B$$, then $$x\not\in A\cap B$$.
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If $$x\not\in A$$ or $$x\not\in B$$, then $$x\in A\cap B$$.
Explanation
The converse of "If P then Q" is "If Q then P"
Hence,
Option A
How many buses are there for Suryapet from Hyderabad?
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$$7$$
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$$17$$
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$$12$$
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$$15$$
Explanation
Buses which stops at Suryapet to be consider.
Since, 'via' signifies the stoppage.
No. of buses to Suryapet from Hyderabad $$=$$ Buses to Suryapet $$+$$ Buses which have stop at Suryapet $$= 7 + 10 = 17$$.
The converse of "if in a triangle $$ABC, AB>AC$$, then $$\angle C=\angle B$$", is
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lf in a triangle $$ABC, \angle C=\angle B$$, then $$AB>AC$$.
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lf in a triangle$$ABC, AB\not\simeq AC$$, then $$\angle C\not\simeq \angle B$$.
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lf in a triangle $$ABC, \angle C\not\simeq \angle B$$, then $$ AB\not\simeq AC$$.
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lf in a triangle $$ABC, \angle C\not\simeq \angle B$$, then $$AB>AC$$.
Explanation
Take
$$p:AB>AC$$
and $$q: \angle C=\angle B$$
So the given statement is symbolically represented as $$p\rightarrow q$$
Now by definition, Converse of a conditional statement $$p\rightarrow q$$ is
$$q\rightarrow p$$
Thus
$$q\rightarrow p$$ is given by
"If in a $$\triangle ABC, \angle C=\angle B$$ then $$AB>AC$$."
The converse of "If $$x$$ has courage, then $$x$$ will win", is
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If $$x$$ wins, then $$x$$ has courage.
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If $$x$$ has no courage, then $$x$$ will not win.
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If $$x$$ will not win, then $$x$$ has no courage.
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If $$x$$ will not win, then $$x$$ has courage.
Explanation
Take $$p:x$$ has courage
and $$q:x$$ will win
So the given conjugation is $$p\Rightarrow q$$
Now we need to find converse of this.
Be definition, Converse will be $$q\Rightarrow p$$
This is symbolic for "If $$x$$ wins then $$x$$ has courage
The contrapositive of "if $$x$$ has courage then $$x$$ will win", is
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If $$x$$ will in, then $$x$$ has courage.
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If $$x$$ has no courage, then $$x$$ will not win.
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If $$x$$ will not win, then $$x$$ has no courage.
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If $$x$$ will not win, then $$x$$ has courage.
Explanation
Take $$p:x$$ has courage
and $$q:x$$ will win
So the given conjugation is $$p\Rightarrow q$$
Now we need to find Contrapositive of this.
Be definition, Contrapositive of
$$p\Rightarrow q$$ is
$$\sim q\longrightarrow \sim p$$
$$\sim q:x$$ will not win and
$$\sim p:x$$ has no courage
Thus the Contrapositive
$$\sim q\longrightarrow \sim p$$ is symbolic for "
if $$x$$ will not win, then $$x$$ has no courge"
The contrapositive of "if in a triangle $$ABC, AB > AC$$, then $$\angle C>\angle B$$", is
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lf in a triangle $$ABC, \angle C>\angle B$$, then $$AB>AC$$.
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lf in a triangle $$ABC, AB\ngtr AC$$, then $$\angle C\ngtr \angle B$$.
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lf in a triangle $$ABC, \angle C\ngtr \angle B$$, then $$AB > AC$$.
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lf in a triangle $$ABC, \angle C\ngtr \angle B$$, then $$ AB\ngtr AC$$ .
Explanation
Take
$$p:AB>AC$$
and $$q:\angle C>\angle B$$
So the given statement is symbolically represented as $$p\rightarrow q$$
Now by definition, Contrapositive of a conditional statement $$p\rightarrow q$$ is $$\sim q\rightarrow \sim p$$
$$\sim p:AB\ngtr AC$$
$$\sim q:\angle C\ngtr \angle B$$
Thus
$$\sim q\rightarrow \sim p$$ is given by
"If in a $$\triangle ABC, \angle C\ngtr \angle B$$ then $$AB\ngtr AC$$."
Here are some words translated from an artificial language
$$ mie \ pie \ is \ blue \ light$$
$$ mie \ tie \ is \ blue \ berry$$
$$ aie \ tie \ is \ rasp \ berry$$
Which words could possibly mean "light fly"?
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pie zie
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pie mie
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aie zie
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aie mie
Explanation
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.
Hence, correct option is pie zie.
Denial of a statement is called its
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negation
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converse
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inverse
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truth value
Explanation
It is a fundamental concept that, denial of a mathematical statement is called its negation.
$$(p\wedge\sim p)\wedge(p\vee q)$$ is a
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contradiction
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tautology
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negation
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none
Explanation
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.
$$(\sim p\wedge q)\wedge q$$ is
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a tautology
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a contradiction
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neither a tautology nor a contradiction
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none of these
Explanation
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.
$$\sim(p\wedge q)\equiv$$
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$$\sim p\vee\sim q$$
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$$ p\vee\sim q$$
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$$\sim p\vee q$$
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None
Explanation
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, $$\sim(p\wedge q)\equiv\sim p\vee\sim q$$
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
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6
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8
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11
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15
Explanation
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.
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"?
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The statement is true but its converse is false
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The statement is false but its converse is true
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Both the statement and its converse are false
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Both the statement and its converse are true
Explanation
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.
If statement $$p \rightarrow (q \vee r)$$ is true then the truth values of statements p, q, r respectively
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T, F, T
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F, T, F
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F, F, F
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all of these
Explanation
$$\because p \rightarrow (q \vee r)$$ 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.
Consider the sentence: x<5
Which of the following integers makes this open sentence true?
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4
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5
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6
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none of the above
Explanation
Of the given options only $$4<5$$ ,i.e; option $$A$$ satisfies $$x<5$$
Which of the following statements is logically equivalent to
"If you live in a mansion, then you have a big heating bill."?
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If you have a big heating bill, then you live in a mansion.
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If you do not live in a mansion, then you do not have a big heating bill.
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If you do not have a big heating bill, then you do not live in a mansion.
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None of these
Explanation
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.
Which of the following statements is the inverse
of
"If you do not understand geometry, then you do not know how to reason deductively."?
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If you reason deductively, then you understand geometry.
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If you understand geometry, then you reason deductively.
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If the do not reason deductively, then you understand geometry.
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None of these
Explanation
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.
Which of the following is the converse of the statement: "If x>4 then x+2>5"?
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If x+2<5 then x<4
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If x is not greater than 4 then x+2 is not greater than 5
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If x+2>5 then x>4
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If x+2 is not greater than 5 then x is not greater than 4
Explanation
Converse of "If $$A$$ then $$B$$" is "If $$B$$ then $$A$$". Hence,
Converse of "If $$x>4$$ then $$x+2>5$$" will be "If
$$x+2>5$$ then $$x>4$$"
So, $$C$$ is correct.
When $$y=3$$ which of the following is FALSE?
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$$y$$ is prime and $$ y$$ is odd
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$$y$$ is odd or $$y$$ is even
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$$y$$ is not prime and $$y$$ is odd
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$$y$$ is odd and $$2y$$ is even
Explanation
Remember what is True: $$3$$ is prime, odd and $$2(3)$$ is even
Choice 1 : T and T is TRUE
Choice 2 : T or F is TRUE
Choice 3 : F and T is FALSE
Choice 4 : T and T is TRUE
"If Deb and Sam go to the mall then it is snowing"
Which statement below is logically equivalent?
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If Deb and Sam do not go to the mall then it is not snowing
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If Deb and Sam do not go to the mall them it is snowing
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If it is snowing then Deb and Sam go to the mall
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If it is not snowing then Deb and Sam do not go to the mall
Explanation
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.
So, $$D$$ is correct.
Which of the following statements is the converse
of
"If the moon is full, then the vampires are prowling."?
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If the vampires are prowling, then the moon is full.
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If the moon is not full, then the vampires are prowling
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If the vampires are not prowling, then the moon is not full.
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None of these
Explanation
Converse of "If $$P$$, then $$Q$$" is "If $$Q$$, then $$P$$"
Similarly, option "$$A$$" is converse of the given statement
"If the moon is full, then the vampires are prowling."
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