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Using the Venn diagram shows that $(A - B)$,$(A \cap B)$ and $(B - A)$ are disjoint sets, taking $A = \left\{ {2,4,6,8,10,\left. {12} \right\}} \right.$and $B = \left\{ {3,6,9,12,\left. {15} \right\}} \right.$

In a class of \[120\] students, $30$ study Maths, $40$ study Physics, $45$ study Chemistry, $15$ study Math and Physics, $20$ study Physics and Chemistry, $12$ study Math and Chemistry and $8$ study all the three. How many are

a.Studying at least one of these subjects

b.did not study any of these

c.how many take only one subject

d.how many study exactly $2$ of the three subjects

a.Studying at least one of these subjects

b.did not study any of these

c.how many take only one subject

d.how many study exactly $2$ of the three subjects

If $A$ is any set, then

A) \[A \cup A^{\prime} = \phi \]

B) \[A \cup A^{\prime} = U\]

C) \[A \cap A^{\prime} = U\]

D) None of the above

A) \[A \cup A^{\prime} = \phi \]

B) \[A \cup A^{\prime} = U\]

C) \[A \cap A^{\prime} = U\]

D) None of the above

If \[A \subset B\] then show that \[A \cap B = {\rm A}\] (Use Venn Diagram).

If $M \cup N = N \cup R$ and $M \cap N = N \cap R$ then which of the following is necessarily true?

A. $M = N$

B. $N = R$

C. $M = R$

D. $M = N = R$

A. $M = N$

B. $N = R$

C. $M = R$

D. $M = N = R$

In order that a relation defined on a non- empty set A is an equivalence relation, it is sufficient, if R

1. Is reflexive

2. Is symmetric

3. Is transitive

4. Possesses all the above properties

1. Is reflexive

2. Is symmetric

3. Is transitive

4. Possesses all the above properties

Which of the following sets are equal?

(i) \[A = \{ 1,2,3,4\}\], \[B = \{ 4,3,2,1\}\]

(ii) \[A = \{ 4,8,12,16\}\], \[B = \{ 8,4,16,18\}\]

(iii) \[X = \{ 2,4,6,8\}\] ,\[Y = \ \{\ x:x\ \text{is a positive even integer} \ 0 < x < 10\ \}\]

(iv) \[P = \ \{\ x:x\ \text{is a multiple of}\ 10,\ x \in N\ \}\] , \[Q = \{ 10,15,20,25,30,\ldots\}\]

(i) \[A = \{ 1,2,3,4\}\], \[B = \{ 4,3,2,1\}\]

(ii) \[A = \{ 4,8,12,16\}\], \[B = \{ 8,4,16,18\}\]

(iii) \[X = \{ 2,4,6,8\}\] ,\[Y = \ \{\ x:x\ \text{is a positive even integer} \ 0 < x < 10\ \}\]

(iv) \[P = \ \{\ x:x\ \text{is a multiple of}\ 10,\ x \in N\ \}\] , \[Q = \{ 10,15,20,25,30,\ldots\}\]

Is the Set of odd numbers between \[7\] and \[19\] an empty set?

If \[X = \left\{ {a,\left\{ {b,c} \right\},d} \right\}\], which of the following is a subset of \[X\] \[?\]

A. \[\left\{ {a,b} \right\}\]

B. \[\left\{ {b,c} \right\}\]

C. \[\left\{ {c,d} \right\}\]

D. \[\left\{ {a,d} \right\}\]

A. \[\left\{ {a,b} \right\}\]

B. \[\left\{ {b,c} \right\}\]

C. \[\left\{ {c,d} \right\}\]

D. \[\left\{ {a,d} \right\}\]

Let \[A = \left\{ {1,2,3} \right\}\] and \[B = \left\{ {2,3,4} \right\}\] , then which of the following relations is a function from A to B?

A \[\left\{ {\left( {1,2} \right),\left( {2,3} \right),\left( {3,4} \right),\left( {2,2} \right)} \right\}\]

B \[\left\{ {\left( {1,2} \right),\left( {2,3} \right),\left( {1,3} \right)} \right\}\]

C \[\left\{ {\left( {1,3} \right),\left( {2,3} \right),\left( {3,3} \right)} \right\}\]

D \[\left\{ {\left( {1,1} \right),\left( {2,3} \right),\left( {3,4} \right)} \right\}\]

A \[\left\{ {\left( {1,2} \right),\left( {2,3} \right),\left( {3,4} \right),\left( {2,2} \right)} \right\}\]

B \[\left\{ {\left( {1,2} \right),\left( {2,3} \right),\left( {1,3} \right)} \right\}\]

C \[\left\{ {\left( {1,3} \right),\left( {2,3} \right),\left( {3,3} \right)} \right\}\]

D \[\left\{ {\left( {1,1} \right),\left( {2,3} \right),\left( {3,4} \right)} \right\}\]

Let S = {1, 2, 3, 4}. The total number of unordered pairs of disjoint subsets of S equal to,

$\left( a \right)$ 25

$\left( b \right)$ 34

$\left( c \right)$ 42

$\left( d \right)$ 41

$\left( a \right)$ 25

$\left( b \right)$ 34

$\left( c \right)$ 42

$\left( d \right)$ 41

Find the union of each of the following pairs of sets:

(i) $X = \{ 1,3,5\} Y = \{ 1,2,3\} $

(ii)$A = \{ a,e,i,o,u\} ,B = \{ a,b,c\} $

(iii)$A = \{ x:x$ is a natural number and multiple of $3\} $

$B = \{ x:x$is a natural number less than $6\} $

(iv)$A = \{ x:x$ is a natural number and $1 < x \leqslant 6\} $

$B = \{ x:x$is a natural number and $6 < x < 10\} $

(v) $A = \{ 1,2,3\} ,B = \phi $

(i) $X = \{ 1,3,5\} Y = \{ 1,2,3\} $

(ii)$A = \{ a,e,i,o,u\} ,B = \{ a,b,c\} $

(iii)$A = \{ x:x$ is a natural number and multiple of $3\} $

$B = \{ x:x$is a natural number less than $6\} $

(iv)$A = \{ x:x$ is a natural number and $1 < x \leqslant 6\} $

$B = \{ x:x$is a natural number and $6 < x < 10\} $

(v) $A = \{ 1,2,3\} ,B = \phi $

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