Complement

Q. If U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}. Verify that (i) (ii)

Q.

If $A=\{x:xisamultipleof4\}andB=\{x:xisamultipleof6\},$ then $A\cap B$ consists of all multiples of

Q.

$\text{A}$ and $\text{B}$ throw a die alternatively till one of them gets a $6$ and wins the game. Find their respective probabilities of winning, if $\text{A}$ starts first .

Q.

If U = {1, 2, 3, .. , 15}, A = {2, 4}, B = {3, 4, 6, 10, 12, 15}, then (A U C)^c is:

1. {1, 2, 3, 5, 15}

2. {1, 5, 7, 8, 9, 11, 13, 14}

3. {3, 4, 6, 10, 12, 15}

4. {1, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}

Q. If $\frac{1}{6!}+\frac{1}{7!}=\frac{x}{8!}$, find $x$.

Q.

If $A=\{x:xisamultipleof3\}andB=\{x:xisamultipleof5\}.$ Then, $A\cap B$ is given by

Q. Let U be the set of all triangles in a plane. If A is the set of all triangles with at least one angle different from 60°, what is ?

Q.

For universal set U, and sets A, B which are subsets of U, the following information is given

n(U) = 47

n(A) = 18

n(B) = 11, n(A ∩ B) = 10

Then, the number of elements that are neither in A nor B is__

Q. In a survey of $600$ students in a school, $150$ students were found to be taking tea and $225$ taking coffee, $100$ were taking both tea and coffee. Find how many students were taking neither tea nor coffee?

Q.

If $U=\left\{x:xisanaturalnumberlessthan20\right\}$, then find its subsets P such that

$P=\left\{evennaturalnumbers\right\}$

Q.

If A = {2, 4, 6, 8} and U = {1, 2, 3, 4, 5, 6, 7, 8, 9, }, then A'=

1. {1, 3, 5, 7, 9}

2. {1, 3, 5, 6, 9}

3. {1, 3, 5, 7}

4. {1, 3, 5, 8, 9}

Q. Evaluate the Given limit: $\underset{x\to \pi }{lim}(x-\frac{22}{7})$

Q.

If A, B and C be the sets such that $A\cup B=A\cup C$ and $A\cap B=A\cap C$ then prove that B= C

Q. Taking the set of natural numbers as the universal set, write down the complements of the following sets: (i) { x : x is an even natural number} (ii) { x : x is an odd natural number} (iii) { x : x is a positive multiple of 3} (iv) { x : x is a prime number} (v) { x : x is a natural number divisible by 3 and 5} (vi) { x : x is a perfect square} (vii) { x : x is perfect cube} (viii) { x : x + 5 = 8} (ix) { x : 2 x + 5 = 9} (x) { x : x ≥ 7} (xi) { x : x ∈ N and 2 x + 1 > 10}

Q. Let U ={1, 2, 3; 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, B = {2, 4, 6, 8} and C = {3, 4, 5, 6}. Find (i) (ii) (iii) (iv) (v) (vi)

Q. Evaluate the given limit: $\underset{x\to 0}{lim}\frac{\cos x}{\pi -x}$

Q. The set $(A\cap B^{\prime }{)}^{\prime }\cup (B\cap C)$ is equal to

1. $A^{\prime }\cup B\cup C$
2. $A^{\prime }\cup B$
3. $A^{\prime }\cup C^{\prime }$
4. $A^{\prime }\cap B$

Q. Evaluate $\underset{x\to 0}{lim}f\left(x\right),$ where $f\left(x\right)=\{\begin{array}{cc}\frac{\left|x\right|}{x},& x\ne 0\\ 0,& x=0\end{array}$

Q. Let $a_1,a_2,\cdot \cdot \cdot \cdot ,a_n$ be fixed real numbers and define a function
$f\left(x\right)=(x-a_1)(x-a_2)$.....$(x-a_n)$
What is $\underset{x\to a_1}{lim}f\left(x\right)?$ For some $a\ne a_1,a_2\cdot \cdot \cdot ,a_n,$ compute $\underset{x\to a}{lim}f\left(x\right)\cdot$

Q.

For universal set U, and sets A, B which are subsets of U, the following information is given -

n(U) = 47

n(A) = 18

n(B) = 11

n(A ∩ B) = 10

Then, the number of elements that neither in A nor B are __

Q.

Let A and B be sets. If $A\cap X=B\cap X=\Phi$ and $A\cup X=B\cup X$ for some set X. Show that $A=B.$

Q.

If A = [ x:x is a multiple of 3] and B = [x:x is a multiple of 5] , then A - B is ($\overline{A}$ means complement of A)

1. 

2. 

3. 

4. 

Q.

In a survey of $600$ students in a school, $150$ students were found to be taking tea and $225$ taking coffee, $100$ were taking both tea and coffee. Find how many students were taking neither tea nor coffee?

Q.

${\left[\right(B^{\prime }\cup (B^{\prime }-A)\left)\right]}^{\prime }$ =___

1. A

2. B

3. A - B

4. B - A

Q. Evaluate the Given limit: $\underset{x\to 1}{lim}\frac{a{x}^2+bx+c}{c{x}^2+bx+a},a+b+c\ne 0$

Q. $\text{If}n(\cup )=700,n\left(A\right)=200,n\left(B\right)=300$ and $n(A\cap B)=100,$ then $n(A^{{}^{\prime }}\cap B^{{}^{\prime }})=$ ___.

1. 400

2. 240

3. 300

4. 500

Q. Fill in the blanks to make each of the following a true statement:
(i) $A\cup A^{\prime }=...$ (ii) ${\varphi }^{\prime }\cap A=...$ (iii) $A\cap A^{\prime }=...$ (iv) $U^{\prime }\cap A=...$

Q. If U = { a, b, c, d, e, f, g, h }, find the complements of the following sets: (i) A = { a, b, c } (ii) B = { d, e, f, g } (iii) C = { a, c, e, g } (iv) D = { f , g , h , a }

Q. For any two complex numbers $z_1$ and $z_2$, prove that $Re\left(z_1{z}_2\right)=Re{z}_{1}Re{z}_2-Im{z}_{1}Im{z}_2$

Q. If A and B are two sets such that $A\subset B$, then what is $A\cup B$?