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CBSE Questions for Class 11 Commerce Applied Mathematics Set Theory Quiz 11 - MCQExams.com
CBSE
Class 11 Commerce Applied Mathematics
Set Theory
Quiz 11
Number of functions from Set-A containing $$5$$ elements to a set-$$B$$ containing $$4$$ elements is
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$${5}^{4}$$
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$${4}^{5}$$
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$$4!$$
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$$5!$$
For any set $$A$$, if $$A\subseteq \phi \Leftrightarrow A=\phi$$.
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True
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False
Explanation
True
Possible sunsets of $$\phi={\phi}$$
$$A\subseteq \phi$$
$$\rightarrow A=\phi$$
If $$A=\left\{ 1,2,4 \right\} ,B=\left\{ 2,4,5 \right\} $$ and $$C=\left\{ 2,5 \right\} $$, then $$\left( A-B \right) \times \left( B-C \right) =$$
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$$\left\{ \left( 1,2 \right) ,\left( 1,5 \right) ,\left( 2,5 \right) \right\} $$
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$$\left\{ \left\{ 1,4 \right\} \right\} $$
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$$\left( 1,4 \right) $$
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$$\left\{ \left( 1,2 \right) \right\} $$
Explanation
$$A=\left\{1,2,4\right\}$$ 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 $$C=\left\{2,5\right\}$$
$$B-C=\left\{2,4,5\right\}-\left\{2,5\right\}=\left\{4\right\}$$
$$\left(A-B\right)\times\left(B-C\right)=\left\{1\right\}\times \left\{4\right\}=\left(1,4\right)$$
if $$A=\left\{x:x^{2}=1\right\}$$ and $$B=\left\{x:x^{4}=1\right\}$$, then $$A \Delta B$$ is equal to
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$$\left\{i,-i\right\}$$
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$$\left\{-1-1\right\}$$
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$$\left\{1-1,i,-i\right\}$$
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$$\left\{1,i\right\}$$
Explanation
Given: $$A=\{x:x^2=1 \}$$ and $$B=\{x:x^4=1 \}$$
Consider, $$A=\{x:x^2=1 \}$$
$$\Rightarrow x^2=1$$
$$\Rightarrow x=\pm1$$
So, $$A=\{-1,1 \}$$
and $$B=\{x:x^4=1 \}$$
$$\Rightarrow x^2=\pm1$$
$$\Rightarrow x^2=1$$ or $$x^2=-1$$
$$\Rightarrow x=\pm1$$ or $$x=\pm-i$$
So, $$B=\{-1,1,-i,i \}$$
Now $$A\Delta B=(A-B)\cup(B-A)=\phi\cup\{i,-i \}$$
So, $$\text{A}$$ is correct option.
{$$x \epsilon R : \dfrac{14x}{x+1} - \dfrac{9x-30}{x-4} <0$$ } is equal to
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$$(-1, 4)$$
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$$(1, 4) \cup (5, 7)$$
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$$(1, 7)$$
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$$(-1, 1) \cup (4, 6)$$
Explanation
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,
$$-1<x<1\quad \mathrm{or}\quad \:4<x<6$$
$$\begin{bmatrix}\mathrm{Solution:}\:&\:-1<x<1\quad \mathrm{or}\quad \:4<x<6\:\\ \:\mathrm{Interval\:Notation:}&\:\left(-1,\:1\right)\cup \left(4,\:6\right)\end{bmatrix}$$
$$n(A \cap B)=2x$$. If $$n(A)=2(n(B))$$ then $$'x'$$ is
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$$4$$
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$$5$$
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$$6$$
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$$7$$
Let A and B be two sets such that $$\\ A\times B$$ has 6 elements. If three elements of $$\\ A\times B$$ are $$\\ \left\{ \left( 1,4 \right) ,\left( 2,6 \right) ,\left( 3,6 \right) \right\} $$, then
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$$A=\left\{ 1,2 \right\} $$ and $$B=\left\{ 3,4,6 \right\} $$
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$$A=\left\{ 4,6 \right\} $$ and $$B=\left\{ 1,2,3 \right\} $$
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$$A=\left\{ 1,2,3 \right\} $$ and $$B=\left\{ 4,6 \right\} $$
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$$A=\left\{ 1,2,4 \right\} $$ and $$B=\left\{ 3,6 \right\} $$
If $$a * b = 2 ^ { a b }$$ on $$N \cup \{ 0 \}$$ then $$*$$ is
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commutative
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associate
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$$(a) \& (b)$$ both
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none of these
If $$A = \{ 2,3,4,5,7 \} , B = \{ 7,8,9 \}$$, then find $$n ( A \cup B ).$$
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$$1$$
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$$3$$
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$$5$$
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$$7$$
Explanation
$$A=\left \{ 2,3,4,5,7 \right \}$$
$$n(A)=5$$
$$B=\left \{ 7,8,9 \right \}$$
$$n(B)= 3$$
$$n(A \cap B)=1$$
$$\therefore (A\cup B)=n(A)+n(B)-n(A\cap B)$$
$$(A\cup B)=5+3-1=7$$
If $$I$$ is the set of isosceles triangle and $$E$$ is the equilateral triangles then _____________.
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$$I\subset E$$
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$$E \subset I$$
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$$E=I$$
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None of these
Explanation
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 $$E\subset I$$.
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Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
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Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
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Assertion is correct but Reason is incorrect
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Both Assertion and Reason are incorrect
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?
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$$126$$
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$$220$$
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$$280$$
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$$144$$
Let $$A, B$$ and $$C$$ be sets such that $$\phi = A\cap B \subseteq C$$. Then which of the following statements is not true?
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If $$(A - C) \subseteq B$$, then $$A\subseteq B$$
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$$(C \cup A)\cap (C\cup B) = C$$
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If $$(A - B)\subseteq C$$, then $$A\subseteq C$$
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$$B\cap C \neq \phi$$
Explanation
for $$A = C, A - C = \phi$$
$$\Rightarrow \phi \subseteq B$$
But $$A\not {\subseteq} B$$
$$\Rightarrow$$ option $$1$$ is NOT true
Let $$x\epsilon (C \cup A)\cap (C\cup B)$$
$$\Rightarrow x\epsilon (C\cup A)$$ and $$x \epsilon (C\cup B)$$
$$\Rightarrow (x \epsilon C$$ or $$x \epsilon A)$$ and $$(x\epsilon C$$ or $$x \epsilon B)$$
$$\Rightarrow x \epsilon C$$ or $$x \epsilon (A\cap B)$$
$$\Rightarrow x \epsilon C$$ or $$x\epsilon C$$ (as $$A\cup B\subseteq C)$$
$$\Rightarrow x \epsilon C$$
$$\Rightarrow (C \cup A)\cap (C\cup B)\subseteq C....(1)$$
Now $$x \epsilon C\Rightarrow x \epsilon (C\cup A)$$ and $$x \epsilon (C \cup B)$$
$$\Rightarrow x\epsilon (C\cup A)\cap (C\cup B)$$
$$\Rightarrow C\subseteq (C\cup A)\cap (C \cup B) .....(2)$$
$$\Rightarrow$$ from (1) and (2)
$$C = (C\cup A)\cap (C\cup B)$$
$$\Rightarrow$$ option 2 is true
Let $$x \epsilon A$$ and $$x \not {\epsilon} B$$
$$\Rightarrow x \epsilon (A - B)$$
$$\Rightarrow x \epsilon C$$ (as $$A - B \subseteq C)$$
Let $$x \epsilon A$$ and $$x \epsilon B$$
$$\Rightarrow x \epsilon (A\cap B)$$
$$\Rightarrow x \epsilon C$$ (as $$A\cap B\subseteq C)$$
Hence $$x \epsilon A \Rightarrow x \epsilon C$$
$$\Rightarrow A \subseteq C$$
$$\Rightarrow$$ Option 3 is true
as $$C\supseteq (A\cap B)$$
$$\Rightarrow B\cap C\supseteq (A\cap B)$$
as $$A\cap B\neq \phi$$
$$\Rightarrow B\cap C \neq \phi$$
$$\Rightarrow$$ Option 4 is true.
State whether the following statement is true or false. Give reason to support your answer.
Every subset of a finite set is finite.
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True
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False
If $$u=\{2, 3, 5, 7, 9\}$$ is the universal set and $$A=\{3, 7\}, B=\{2, 5, 7, 9\}$$, then find the following statement is true/false.
$$(A\cap B)'=A'\cap B'$$.
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True
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False
Explanation
The given sets are:
$$u=\left\{2,3,5,7,9 \right\}\ A=\left\{3,7 \right\}\ B=\left\{2,5,7,9 \right\}$$
$$A\cap B=\left\{7 \right\}$$
$$(A\cap B)'=u-(A\cap B)=\left\{2,3,5,9 \right\}$$
$$A'=u-A=\left\{2,5,9 \right\}$$
$$B'=u-B=\left\{3 \right\}$$
$$A' \cap B'=\phi$$
$$\Rightarrow \ \boxed {(A\cap B)' \neq A' \cap B'}$$
Mark the correct alternative of the following.
The number of subsets of a set containing n elements is?
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n
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$$2^n-1$$
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$$n^2$$
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$$2^n$$
State whether the following statement is true or false. If the statement is false, re-write the given statement correctly.
If $$A=\{1, 2\}, B=\{3, 4\}$$, then $$A\times (B\cap \phi)=\phi$$.
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True
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False
State whether the following statement is true or false. If the statement is false, re-write the given statement correctly.
If $$P=\{m, n\}$$ and $$Q=\{n, m\}$$, then $$P\times Q=\{(m, n), (n, m)\}$$.
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True
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False
Examine whether the following statements are true or false:
$$a\in \left\{\{a\}\, b \right\}$$
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True
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False
Examine whether the following statements are true or false:
$$(a,e) \subset$$ ($$x : x$$ is a vowel in the English alphabet)
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True
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False
Explanation
True, $$a,e$$ are two vowels of the English alphabet.
Examine whether the following statements are true or false:
($$x : x$$ is an even natural number less than $$6$$ ) $$\subset $$ ($$x: x$$ is a natural number which divide $$36$$.
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True
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False
Explanation
True. ($$x : x$$ is an even natural number less than $$6$$ )= $$(2,4)$$
($$x : x$$ is a natural number which divides $$36$$ )= $$(1,2,3,4,6,9,12,18,36)$$.
Examine whether the following statements are true or false:
$$(a)\subset (a,b,c)$$
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True
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False
Explanation
True. Each element of $$(a)$$ is also an element of $$(a, b, c)$$.
Examine whether the following statements are true or false:
$$(a,b)\not{\subset}(b,c,a)$$
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True
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False
Explanation
False. Each element of $$(a,b)$$ is also an element of $$(b,c,a)$$.
In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.
If $$x A$$ and $$A B$$, then $$x B$$
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True
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False
Explanation
False
Let $$A = \left \{ 1,2 \right \}$$ and $$B = \left \{ 1,\left \{ 1,2 \right \} ,\left \{ 3 \right \}\right \}$$
Now, $$2 ∈ \left \{ 1,2 \right \}$$ and $$\left \{ 1,\left \{ 1,2 \right \} ,\left \{ 3 \right \}\right \}$$
$$\therefore A ∈ B$$
However, $$2 ∉ \left \{ \left \{ 3 \right \},1,\left \{ 1,2 \right \} \right \}$$.
In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.
If $$A B$$ and $$B C$$, then $$A C$$
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True
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False
Explanation
False
Let $$A=\left \{ 1,2 \right \},B=\left \{ 0,6,8 \right \}$$ and $$C=\left \{ 0,1,2,6,9 \right \}$$
Accordingly $$A ⊄ B$$ and $$B ⊄ C$$
However, $$A ⊂ C$$.
In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.
If $$A B$$ and $$x B$$, then $$x A$$
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True
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False
Explanation
True
Let, $$A ⊂ B$$ and $$x ∉ B$$
To show: $$x \notin A$$,
If possible, suppose $$x ∈ A$$,
Then, $$x\in B$$, which is a contradiction as $$x ∉ B$$
$$\therefore x ∉ A$$
Choose the correct answer from the given four options
Which of the following collection doesn't form a set
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Collection of 5 odd prime numbers
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Collection of 3 most intelligent students of your class
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Collection of 4 vowels of the English alphabet
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Collection of first 6 months of a year
Explanation
Collection of 5 odd prime number. Collection of 4 vowels of English alphabet and collection of first 6 months of a year, all are sets but a collection of 3 most intelligent students of your class is not a set, because intelligence is not well defined.(b)
State whether the following statements are true(T) or false(F):
A collection of books is a set.
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True
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False
Explanation
A collection of books is a set.(False)
Correct:
A collection of different books is a set.
Choose the correct answer from the given four options
If A = {x | x is a positive multiple of 3 less than 20} and B = {x | x is a prime number less than 20}, then n(A) + n(B) is
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$$6$$
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$$8$$
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$$13$$
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$$14$$
Explanation
If$$ A = \{x | x$$ is a positive multiple of $$3$$ less than $$20\}$$
$$=\left\{3,6,9,12,15,18\right\}\Rightarrow n(A)=6$$
$$B = \{x | x$$ is a prime number less than $$20\}$$
$$=\left\{2,3,5,7,11,13,17,19\right\} \Rightarrow n(B)=8$$
$$n(A) + n(B) = 6+8=14$$
State true or false for each of the following. Correct the wrong statement If $$A = \{0\}$$, then $$n (A) = 0$$
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True
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False
Explanation
Given
If $$A = \{0\}$$, then $$n (A) = 0$$
The statement given here is false
Correct statement: If $$A = \{0\}$$, then $$n (A) = 1$$
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Practice Class 11 Commerce Applied Mathematics Quiz Questions and Answers
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