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CBSE Questions for Class 11 Engineering Maths Mathematical Reasoning Quiz 4 - MCQExams.com
CBSE
Class 11 Engineering Maths
Mathematical Reasoning
Quiz 4
State the following statement is True or False
If $$p, q, r$$ are statements with truth values T, F, T respectively, then the truth value of $$(\sim p\vee q)\wedge \sim r \rightarrow p$$, is T
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True
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False
Explanation
$$p$$
$$q$$
$$r$$
$$\sim p\vee q$$
$$\left( \sim p\vee q \right) \wedge \sim r$$
$$\left( \sim p\vee q \right) \wedge \sim r\rightarrow p$$
T
F
T
F
F
T
The contrapositive of $$(p \vee q)\Rightarrow r$$ is
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$$r \Rightarrow (p \vee q)$$
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$$\sim r \Rightarrow (p \vee q)$$
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$$\sim r \Rightarrow \sim p \wedge \sim q$$
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$$R \Rightarrow (q \vee r)$$
Explanation
contrapositive of $$a \implies b $$ is $$ \sim b \implies \sim a$$
contrapositive of $$ p \lor q \implies r $$ is
$$\sim r \implies \sim ( p \lor q )$$
$$ \sim r \implies (\sim p \land \sim q )$$
Which of the following is/are a statement?
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Give me a glass of water.
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Asia is continent.
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The earth revolves round sun.
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The number $$6$$ has two prime factor $$2,3$$.
Explanation
In mathematics, A sentence is called 'a statement' if it is either correct or incorrect but not both.
Verify options.
State the following statement is True or False
If $$p, q, r$$ are statements, with truth values T, F, T respectively, then the truth value of $$(\sim p \vee q) \wedge \sim r \Rightarrow p$$ is T
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True
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False
Explanation
$$p,q,r$$ are statements with truth values $$T,F,T$$ respectively
Consider
$$(\sim p \vee q) \wedge \sim r \Rightarrow p$$
$$=(F \vee F) \wedge F \Rightarrow T$$
$$=(F \wedge F) \Rightarrow T$$
$$=F \Rightarrow T=\sim(F) \vee T=T\vee T=T$$
Clearly, option A.
If $$p\rightarrow (q \vee r)$$ is false, then the truth values of $$p,q,r$$ are respectively
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T, F, F
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F, F, F
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F, T, T
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T, T, F
Explanation
$$p \rightarrow q$$ is false only when $$p$$ is true and $$q$$ is false.
$$\therefore p\rightarrow (q \vee r)$$ is false when $$p$$ is true and $$(q \vee r)$$ is false, and
$$q \vee r$$ false when both $$q, r$$ are false.
Hence T,F,F
If $$x = 5$$ and $$y = -2$$, then $$x-2y = 9$$. The contrapositive of this statement is/are
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If $$x-2y$$ $$\neq 9$$, then $$x \neq$$ $$5$$ or $$y \neq 2$$.
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If $$x-2y$$ $$\neq 9$$, then $$x \neq 5$$ and $$y \neq -2$$.
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If $$x-2y = 9$$, then $$x = 5$$ and $$y = -2$$.
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none of these.
Explanation
Let $$p,q,r$$ be the three statement.
$$ p : x = 5,\>q : y = 2,\>r : x - 2y = 9$$
Here given statement is $$(p \wedge q) \rightarrow r$$ and its contrapositive is
$$\sim r \rightarrow \sim (p \wedge q)$$
i.e. $$\sim r \rightarrow (\sim p \vee \sim q)$$
i.e. If $$x - 2y \neq 9$$, then $$x \neq 5$$ or $$y \neq 2$$
The negation of the statement: "If I become a teacher, then I will open a school" is
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I will become a teacher and I will not open a school.
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Either I will not become a teacher or I will not open a school.
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Neither I will become a teacher nor I will open a school.
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I will not become a teacher or I will open a school.
Explanation
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'.
If $$x = 5$$ and $$ y = 2$$ then $$x - 2y = 9$$. The contrapositive of this statement is
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If $$x -2y \neq 9$$ then $$x \neq 5$$ or $$y \neq 2$$
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If $$x -2y \neq 9 $$ then $$x \neq 5$$ and $$y \neq - 2$$
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If $$x - 2y = 9$$ then $$x = 5$$ and $$y = -2$$
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none of these.
Explanation
given three statements p: $$x=5$$
q : $$y=2$$
r: $$x+2y=9$$
$$(p \land q ) \implies r $$ contrapositive of this is
$$\sim r \implies \sim (p \land q)$$
$$\therefore x+2y \ne 9 \implies x \ne 5$$ or $$ y\ne 2 $$
Choose the conclusion of given statements:
All scientists working in America are talented. Some Indian scientists are working in America. Therefore, "Some Indian scientists are talented."
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True
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May be true
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False
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May be false
Explanation
The statement is true as it's already given that
All scientists working in America are talented.
Consider the statements
(i)Two plus three is five.
(ii) Every square is a rectangle.
(iii) Sun rises in the east.
(iv) The earth is not a star.
Which of the above statements have truth value (T) ?
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(i) and (ii)
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(ii) and (iii)
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(iii) and (iv)
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All of these
Explanation
We know, If a statement is true then its truth value is T and if statement is false then F.
Hence all the statements (i),(ii),(iii) and (iv) are true hence their truth value is T
The contrapositive of $$p\rightarrow (\sim q\rightarrow \sim r)$$ is equivalent to
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$$(\sim q\wedge r)\rightarrow \sim p$$
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$$(q\wedge \sim r)\rightarrow \sim p$$
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$$p\rightarrow (\sim r\vee q)$$
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$$p\wedge (q\vee r)$$
Explanation
The contrapositive of $$p \rightarrow ( \sim q \rightarrow \sim r)$$ is
$$ \equiv \sim (\sim q \rightarrow \sim r) \rightarrow \sim p $$
$$ \equiv \sim ( q \vee \sim r) \rightarrow p \equiv (\sim q \wedge r) \rightarrow \sim p $$
Which one of the following statements is not a false statement?
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$$p:$$ Each radius of a circle is a chord of the circle.
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$$q:$$ Circle is a particular case of an ellipse.
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$$r:\>\displaystyle \sqrt{13}$$ is a rational number.
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$$s:$$ The centre of a circle bisects each chord of the cirlce.
Explanation
We know that equation of an ellipse is given by $$\displaystyle \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$$
If we take $$\displaystyle a=b $$ then we get $$\displaystyle x^{2}+y^{2}=a^{2} $$ which satisfies all the conditions of circle
$$\displaystyle \therefore $$ circle is the particular case of an ellipse.
Ans: B
Consider the. following compound statement
(i) Mumbai is the capital of Rajasthan or Maharashtra,
(ii) $$\displaystyle \sqrt{3}$$ is a rational number or an irrational number,
(iii) $$125$$ is a multiple of $$7$$ or $$8$$
(iv) A rectangle is a quadrilateral or a regular hexagon.
Which of the above statements is not true?
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(i)
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(ii)
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(iii)
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(Iv)
Explanation
(i) The component statements of " Mumbai is the capital of Rajasthan or Maharashtra" are
$$p :$$ Mumbai is the capital of Rajasthan.
$$q :$$ Mumbai is the capital of Maharashtra.
We note that $$p$$ is false and $$q$$ is true, so the compound statement is true
(ii) The component statements of $$\displaystyle \sqrt{3}$$ is a rational or an irrational are
$$p :$$ $$\displaystyle \sqrt{3}$$ is a rational number.
$$q:$$ $$\displaystyle \sqrt{3}$$ is an irrational number.
We note that $$p$$ is false and $$q$$ is true, so the compound statement is true.
(iii) The component statements of $$125$$ is a multiple of $$7$$ or $$8$$ are
$$p:125$$ is a multiple of $$7.$$
$$q:125$$ is a multiple of $$8.$$
We note that $$p$$ and $$q$$ both are false statements, so compound statement is false.
(iv) The component statements of "A rectangle is a quadrilateral or a regular hexagon." are
$$p: A$$ rectangle is a quadrilateral.
$$q: A$$ rectangle is a regular hexagon.
We note that $$p$$ is true and $$q$$ is false, so the compound statement is true.
The contrapositive of $$(\sim p\wedge q)\rightarrow \sim r$$ is equivalent to
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$$(p\wedge q)\rightarrow r$$
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$$(p\wedge q)\vee r$$
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$$r\rightarrow (p\vee \sim q)$$
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none of these
Explanation
The contrapositive of $$(\sim p\wedge q)\rightarrow \sim r$$ is,
$$\equiv \sim (\sim r) \rightarrow \sim (\sim p\wedge q)$$
$$\equiv r \rightarrow ( p \vee \sim q)$$
The contrapositive of the statement "If you believe in yourself and are honest then you will get sucess" is
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If you do not believe yourself and are dishonest then you will not get success.
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If you do not believe yourself and are dishonest then you will get success.
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If you get success then you are honest and you also believe in yourself.
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If you will not get success then you don't not believe in yourself or are not honest
Explanation
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.
The converse of $$p \rightarrow (q \rightarrow r)$$ is
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$$(q \wedge \sim r) \vee p $$
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$$ (\sim q \vee r)\vee p$$
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$$(q \wedge \sim r) \wedge \sim p $$
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$$(q \wedge \sim r) \wedge p $$
Explanation
The converse of $$p \rightarrow (q \rightarrow r)$$ is,
$$\equiv (q \to r) \to p \equiv (\sim q \vee r) \to p \equiv (q\wedge \sim r) \to p$$
In the following letter sequence, some of the letters are missing. These are given in order as one of the alternatives below. Choose the correct alternative.
$$\alpha \beta$$ _$$\alpha \alpha$$ _$$\beta \beta \beta$$ _$$\alpha \alpha \alpha \alpha$$ _$$\beta \beta \beta ...$$
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$$\alpha \beta \beta \alpha $$
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$$\beta \alpha \beta \alpha $$
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$$\alpha \alpha \alpha \beta $$
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$$\alpha \beta \alpha \beta $$
Explanation
This follows the following pattern
$$\alpha |\beta \beta |\alpha \alpha \alpha |\beta \beta \beta \beta |\alpha \alpha \alpha \alpha \alpha |\beta \beta \beta \beta \beta \beta $$
Therefore $$\beta\alpha\beta\alpha$$ is the missing part.
Hence, option 'B' is correct..
The negation of the statement $$q \vee (p \wedge \sim r)$$ is equivalent to
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$$\sim q \wedge(p \rightarrow r)$$
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$$\sim q \vee \sim(p \rightarrow r)$$
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$$q \wedge (\sim p \wedge r)$$
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None of these.
Explanation
The negation of the statement $$q \vee (p \wedge \sim r) $$ is
$$\equiv \sim (q \vee (p \wedge \sim r) ) \equiv \sim q \wedge (\sim p \vee \sim(\sim r))\equiv \sim q \wedge (\sim p \vee r)\equiv \sim q \wedge (p\to r) $$
The converse of $$p \rightarrow (q \rightarrow r)$$ is
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$$(q \wedge \sim r) \vee p$$
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$$(\sim q \vee r) \vee p$$
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$$(q \wedge \sim r) \wedge \sim p$$
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$$(q \wedge \sim r) \wedge p$$
Explanation
The converse of $$p \rightarrow (q \rightarrow r)$$ is,
$$\equiv (q \to r) \to p \equiv (\sim q \vee r) \to p \equiv \sim (\sim q \vee r) \vee p\equiv (q \wedge \sim r) \vee p $$
The contrapositive of $$\sim p \rightarrow ( q \rightarrow \sim r)$$ is
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$$(~q \wedge r) \rightarrow ~p$$
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$$(q \rightarrow r) \rightarrow~p$$
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$$(q \vee ~r) \rightarrow ~ p$$
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None of these
Explanation
The contrapositive of $$\sim p \rightarrow ( q \rightarrow \sim r)$$ is
$$\sim ( q \rightarrow \sim r) \rightarrow \sim (\sim p )$$
$$\equiv \sim (\sim q \vee \sim r) \rightarrow p \equiv (q \wedge r) \rightarrow p $$
The negative of the statement "If a number is divisible by $$15$$ then it is divisible by $$5$$ or $$3$$"
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If a number is not divisible by $$15$$, then it is not divisible by $$5$$ and $$3$$
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A number is divisible by $$15$$ and it is not divisible by $$5$$ or $$3$$
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A number is not divisible by $$15$$ or it is not divisible by $$5$$ and $$3$$
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A number is divisible by $$15$$ and it is not divisible by $$5$$ and $$3$$
Explanation
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 $$p \rightarrow (q \vee r)$$
Here negative of above statement is
$$\sim (p \rightarrow (q \vee r))\equiv p \wedge (\sim (q \vee r)$$
$$\equiv p \wedge ( \sim q \wedge \sim r)$$
i.e. A number is divisible by $$15$$ and it is not divisible by $$5$$ and $$3$$.
If statements $$p, q, r$$ have truth values T, F, T respectively then which of the following statement is true
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$$(p \rightarrow q) \wedge r $$
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$$ (p \rightarrow q) \vee \sim r$$
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$$(p \wedge q)\vee (q \wedge r) $$
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$$ (p \rightarrow q) \rightarrow r$$
Explanation
A. $$(p \rightarrow q) \wedge r \equiv (T \rightarrow F) \wedge T \equiv F \wedge T \equiv F$$
B. $$(p \rightarrow q) \vee \sim r\equiv (T \rightarrow F) \vee F \equiv F \vee F \equiv F$$
C. $$(p \wedge q) \vee (q \wedge r)\equiv(T \wedge F) \vee (F \wedge T)\equiv F\vee F \equiv F $$
D. $$(p \rightarrow q) \rightarrow r \equiv (T \rightarrow F) \rightarrow T\equiv F\rightarrow T \equiv T$$
The contrapositive of $$p \rightarrow (\sim q \rightarrow \sim r)$$ is
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$$(\sim q\wedge r) \rightarrow \sim p $$
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$$ (q \rightarrow r) \rightarrow \sim p$$
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$$( q \vee \sim r) \rightarrow \sim p $$
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none of these.
Explanation
The contrapositive of $$p \rightarrow (\sim q \rightarrow \sim r)$$ is
$$\equiv \sim (\sim q \rightarrow \sim r) \rightarrow \sim p$$
$$\equiv \sim(q \vee \sim r) \to \sim p$$
$$\equiv (\sim q \wedge r) \to \sim p$$
Negation of $$\displaystyle q \vee \sim \left ( p\wedge r \right )$$ is
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$$\displaystyle \sim q\wedge \sim \left ( p\wedge r \right )$$
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$$\displaystyle \sim q\wedge \left ( p\wedge r \right )$$
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$$\displaystyle \sim q\vee \left ( p\wedge r \right )$$
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None of these
Explanation
$$\displaystyle \sim \left ( q\vee \sim \left ( p\wedge r \right )
\right ) \equiv \sim q\wedge \left ( \sim \left ( \sim (p\wedge r \right )
\right ) \equiv \sim q\wedge \left ( p\wedge r \right )$$
Find the truth value of the compound statement, 4 is the first composite number and $$2+5=7$$.
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T
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F
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Neither T nor F
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cannot be determined
Explanation
The given statement is True, as $$ 4$$ is the first composite number. Also $$ 2 + 5 = 7 $$
The negation of the statement "$$2 + 3 = 5$$" and "$$8 < 10$$" is
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$$2 + 3 \neq 5$$ and $$8\nless 10$$
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$$2 + 3 $$\neq 5 or $$8 > 10$$
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$$2 + 3 \neq 5$$ or $$8 \ngeqslant 10$$
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None of these
Explanation
Take $$p:2+3=5$$ and $$q:8<10$$
So the given conjunction is $$p\wedge q$$
Now $$\sim p:2+3\neq 5$$ and $$\sim q:8\nless 10$$
Now Negation of the given conjunction
$$p\wedge q$$ is
$$\sim \left( p\wedge q \right) $$
$$\sim \left( p\wedge q \right) :2+3\neq 5$$ or $$8\nless 10$$
Find the quantifier which best describes the variable of the open sentence $$x^2+2\ge0$$
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Universal.
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Existential.
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Neither (a) nor (b).
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Does not exist.
Explanation
An universal quantifier is a symbol or a logic to denote that the statement is true for all values under the scope.
We know that $$ x^2 $$ is always $$ \ge 0 $$
Hence, $$ x^2 + 2 \ge 0 $$ holds true for all values of $$ x $$.
Thus, the quantifier to describe the variable of the given sentence is universal.
The inverse of the statement $$(p \wedge \sim q) \rightarrow r$$ is
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$$\sim (p \vee \sim q) \rightarrow \sim r$$
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$$(\sim p \wedge \sim q) \rightarrow \sim r$$
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$$(\sim p \vee q) \rightarrow \sim r$$
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None of these.
Explanation
$$\textbf{Step-1: Apply the concept of logical reasoning. }$$
$$\text{We have,}$$
$$(p \wedge \sim q) \rightarrow r$$
$$\therefore$$ $$\text{The inverse of the statement}$$ $$(p \wedge \sim q) \rightarrow r$$ $$\text{is,}$$
$$\equiv \sim (p \wedge \sim q) \rightarrow \sim r \equiv (\sim p \vee \sim (\sim q)) \to \sim r$$
$$\equiv (\sim p \vee q) \to \sim r$$
$$\textbf{Hence, option C}$$
What is the truth value of the statement 'Two is an odd number iff 2 is a root of $$x^2+2=0$$'?
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T
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F
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Neither T nor F
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Cannot be determined
Explanation
Both the statements are false as Two is not an odd number and $$ 2 $$ is not the root of $$ x^2 + 2 = 0 $$
We know that two False statements together make a True statement., Hence, truth value of the given statement is True.
The negative of the statement "If a number is divisible by 15 then it is divisible by 5 or 3"
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if a number is divisible by 15 then it is not divisible by 5 and 3
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a number is divisible by 15 and it is not divisible by 5 or 3
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a number is divisible by 15 or it is not divisible by 5 and 3
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a number is divisible by 15 and it is not divisible by 5 and 3
Explanation
Let $$p, q, r$$ be three statements defined as
$$p$$ : a number N is divisible by 15
$$q$$ : a number N is divisible by 5
$$r$$ :a number N is divisible by 3
Here given statement is $$p\rightarrow (q\vee r)$$
Here negative of above statement is
$$\sim (p \rightarrow (q \vee r)) $$
$$=\sim [(\sim p) \vee (q \vee r)] $$ ($$\because p\rightarrow q=\sim p \vee q$$)
$$= \sim (\sim p) \wedge (\sim (q \vee r)$$
$$= p \wedge (\sim q \wedge \sim r)$$ ($$\because \sim (\sim p)=p, \sim (p\vee q=\sim p \wedge \sim q$$)
i.e. a number is divisible by 15 and it is not divisible by 5 and 3.
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