Sets Test

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Sets Test - 42

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Weekly Quiz Competition

Question 1
1 / -0

Let $$p$$ and $$q$$ be two statements. amongst the following, the statement is equivalent to $$p\rightarrow q$$ is

A
$$\displaystyle p\wedge \sim q$$

B
$$\displaystyle \sim p\vee q$$

C
$$\displaystyle \sim p\wedge q$$

D
$$\displaystyle p\vee \sim q$$

Solution

$$ \begin{array}{l} \text { Given, } \quad P \longrightarrow q \\ \text { ne trinow that, } \rightarrow \text { - if then } \\ \Rightarrow \text { if } P \text { then } q \end{array} $$
$$ \begin{array}{lll} p & q & p \rightarrow q \\ 1 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{array} $$
$$ \begin{array}{c} \Rightarrow \text { Evaluating option } A(p \wedge \sim q) \\ P \text { iq } p \wedge \sim q \\ 111 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \\ \text { Incorrect assumption } \end{array} $$
$$ \begin{array}{c} \Rightarrow \text { Eualuating } B(\sim p \wedge q) \\ \sim p \quad q \quad \sim p \wedge q \\ 0 \\ 0 \\ 1 \\ 1 \quad 0 \\ \text { Incorrect assumption } \end{array} $$
$$ \begin{array}{rlrl} \Rightarrow \text { Evaluating } & c(\sim p \vee q) \\ & \sim p & q & \sim p v q \\ & 0 & 1 & & \\ 0 & 0 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \\ & \text { correct assumption } \end{array} $$
Question 2
1 / -0

If $$A=\left\{ 1,3,5,7,9,11,13,15,17 \right\} ,B=\left\{ 2,4,....,18 \right\} $$ and N is the universal set, then $$A'\cup \left( \left( A\cup B \right) \cap B' \right) $$

A
A

B
N

C
B

D
R

Solution

$$A=\left\{1,3,5,7,9,11,13,15,17\right\}$$

$${A}^{\prime}=N-A=N-\left\{1,3,5,7,9,11,13,15,17\right\}=\left\{2,4,6,8,10,12,14,16,18\right\}$$

$$A\cup B=\left\{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18\right\}$$

$${B}^{\prime}=N-B=N-\left\{2,4,6,8,10,12,14,16,18\right\}=\left\{1,3,5,7,9,11,13,15,17\right\}$$

$$\left(A\cup B\right)\cap {B}^{\prime}$$

$$=\left\{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18\right\}\cap\left\{1,3,5,7,9,11,13,15,17\right\}$$

$$=\left\{1,3,5,7,9,11,13,15,17\right\}$$

$${A}^{\prime}\cup \left(\left(A\cup B\right)\cap {B}^{\prime}\right)$$

$$=\left\{2,4,6,8,10,12,14,16,18\right\}\cup\left\{1,3,5,7,9,11,13,15,17\right\}$$

$$=\left\{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18\right\}=N$$
Question 3
1 / -0

If $$a * b = 2 ^ { a b }$$ on $$N \cup \{ 0 \}$$ then $$*$$ is

A
commutative

B
associate

C
$$(a) \& (b)$$ both

D
none of these

Solution
Question 4
1 / -0

If $$A=\left\{ 1,2,4 \right\} ,B=\left\{ 2,4,5 \right\} $$ and $$C=\left\{ 2,5 \right\} $$, then $$\left( A-B \right) \times \left( B-C \right) =$$

A
$$\left\{ \left( 1,2 \right) ,\left( 1,5 \right) ,\left( 2,5 \right) \right\} $$

B
$$\left\{ \left\{ 1,4 \right\} \right\} $$

C
$$\left( 1,4 \right) $$

D
$$\left\{ \left( 1,2 \right) \right\} $$

Solution

$$A=\left\{1,2,4\right\}$$ and $$B=\left\{2,4,5\right\}$$

$$A-B=\left\{1,2,4\right\}-\left\{2,4,5\right\}=\left\{1\right\}$$

$$B=\left\{2,4,5\right\}$$ and $$C=\left\{2,5\right\}$$

$$B-C=\left\{2,4,5\right\}-\left\{2,5\right\}=\left\{4\right\}$$

$$\left(A-B\right)\times\left(B-C\right)=\left\{1\right\}\times \left\{4\right\}=\left(1,4\right)$$
Question 5
1 / -0

{$$x \epsilon R : \dfrac{14x}{x+1} - \dfrac{9x-30}{x-4} <0$$ } is equal to

A
$$(-1, 4)$$

B
$$(1, 4) \cup (5, 7)$$

C
$$(1, 7)$$

D
$$(-1, 1) \cup (4, 6)$$

Solution

Given,

$$\dfrac{14x}{x+1}-\dfrac{9x-30}{x-4}<\:0$$

$$\dfrac{5x^2-35x+30}{\left(x+1\right)\left(x-4\right)}<0$$

$$\dfrac{(x-1)(x-6)}{\left(x+1\right)\left(x-4\right)}<0$$

For $$(x-1),(x-6)$$ and $$(x+1),(x-4)$$ we get,

$$-1<x<1\quad \mathrm{or}\quad \:4<x<6$$

$$\begin{bmatrix}\mathrm{Solution:}\:&\:-1<x<1\quad \mathrm{or}\quad \:4<x<6\:\\ \:\mathrm{Interval\:Notation:}&\:\left(-1,\:1\right)\cup \left(4,\:6\right)\end{bmatrix}$$
Question 6
1 / -0

$$A = \{ a , e , i , o , u \} , B = \{ a , i , u \}$$, then $$B - A.$$

A
$$\{ e , o \}$$

B
$$\{ a , i , u \}$$

C
$$\{ o \}$$

D
$$\{ \}$$

Solution
Question 7
1 / -0

If $$A=\left\{x|x\in N\quad and\quad x < 6\dfrac{1}{4}\right\}$$ and $$B=\left\{x|x\in N\quad and\quad x^2\leq 5\right\}$$. Then the number of subsets of set $$A\times (A\cap B)$$ which contains exactly $$3$$ elements is?

A
$$126$$

B
$$220$$

C
$$280$$

D
$$144$$

Solution
Question 8
1 / -0

If $$A = \{ 2,3,4,5,7 \} , B = \{ 7,8,9 \}$$, then find $$n ( A \cup B ).$$

A
$$1$$

B
$$3$$

C
$$5$$

D
$$7$$

Solution

$$A=\left \{ 2,3,4,5,7 \right \}$$

$$n(A)=5$$

$$B=\left \{ 7,8,9 \right \}$$

$$n(B)= 3$$

$$n(A \cap B)=1$$

$$\therefore (A\cup B)=n(A)+n(B)-n(A\cap B)$$

$$(A\cup B)=5+3-1=7$$
Question 9
1 / -0

If $$I$$ is the set of isosceles triangle and $$E$$ is the equilateral triangles then _____________.

A
$$I\subset E$$

B
$$E \subset I$$

C
$$E=I$$

D
None of these

Solution

Given, $$I$$ is the set of isosceles triangle and $$E$$ is the equilateral triangles.
We know that every equilateral triangle is an isosceles triangle but the converse is not true.
Hence $$E\subset I$$.
Question 10
1 / -0

Mark the correct alternative of the following.
The number of subsets of a set containing n elements is?

A
n

B
$$2^n-1$$

C
$$n^2$$

D
$$2^n$$

Solution

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Re-Attempt Weekly Quiz Competition

Sets Test - 42

Result

Sets Test - 42

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SHARING IS CARING

Question 1

$$\displaystyle p\wedge \sim q$$

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

$$\displaystyle \sim p\wedge q$$

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

Solution

Question 2

A

N

B

R

Solution

Question 3

commutative

associate

$$(a) \& (b)$$ both

none of these

Solution

Question 4

$$\left\{ \left( 1,2 \right) ,\left( 1,5 \right) ,\left( 2,5 \right) \right\} $$

$$\left\{ \left\{ 1,4 \right\} \right\} $$

$$\left( 1,4 \right) $$

$$\left\{ \left( 1,2 \right) \right\} $$

Solution

Question 5

$$(-1, 4)$$

$$(1, 4) \cup (5, 7)$$

$$(1, 7)$$

$$(-1, 1) \cup (4, 6)$$

Solution

Question 6

$$\{ e , o \}$$

$$\{ a , i , u \}$$

$$\{ o \}$$

$$\{ \}$$

Solution

Question 7

$$126$$

$$220$$

$$280$$

$$144$$

Solution

Question 8

$$1$$

$$3$$

$$5$$

$$7$$

Solution

Question 9

$$I\subset E$$

$$E \subset I$$

$$E=I$$

None of these

Solution

Question 10

n

$$2^n-1$$

$$n^2$$

$$2^n$$

Solution

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