Union of Sets

Q. Let A=\{a, b, c, d\}. B and C are two sets such that B⊂eq A, C⊂eq A but B∩ C=ϕ. Number of possible ordered pairs (B, C) satisfying the above conditions is.

Q.

Two events $A$ and$B$ are such that:$P\left(A\right)=\frac{8}{15},P\left(B\right)=\frac{1}{3},P\left(A|B\right)=\frac{1}{5}$

Calculate the probability that:

$a)$Both events occur

$b)$Only one of the two events occurs.

$c)$Neither event occurs

Q.

Two cards are drawn at random from a pack of 52 cards. What is the probability that both the drawn cards are aces ?

Q.

If A = {2, 4} and B = {5, 6} then A x B ∪ ф will be:

1. {(2, 5), (4, 6)}

2. {(2, 5), (2, 6), (4, 5), (4, 6)}

3. ф

4. {2, 4, 5, 6}

Q. Let $A$ and $B$ be two events such that the probability that exactly one of them occurs is $\frac{2}{5}$ and the probability that $A$ or $B$ occurs is $\frac{1}{2}$, then the probability of both of them occur together is :

1. $0.10$
2. $0.20$
3. $0.01$
4. $0.02$

Q.

Let $A$ and $B$ be two events such that the probability that exactly one of them occurs is $\frac{2}{5}$ and the probability that $A$or $B$ occurs is $\frac{1}{2}$, then the probability of both of them occurring together is

1. $0.10$

2. $0.20$

3. $0.01$

4. $0.02$

Q.

For the three events A, B and C, P (exactly one of the events A or B occurs) = P(exactly one of the events B or C occurs) = P(exactly one of the events C or A occurs)=p and P(all the three events occur simultaneously)=$p^2$, where 0<p<1/2. Then the probability of at least one of the three events A, B and C occuring is

1. $\frac{3p+2p^2}{2}$
   

2. $\frac{p+3p^2}{4}$
   

3. $\frac{3p+2p^2}{4}$

4. $\frac{p+3p^2}{2}$
   

Q. A = {1, 2, 3, 6}, B = {2, 3, 5, 7} and U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.
Then, A' ∩ B' =

1. {2, 3}
2. {4, 8, 9, 10}
3. {1, 2, 3, 5, 6, 7}
4. {1, 2, 5, 7}

Q. For any two sets A and B, show that the following statements are equivalent:
(i) $A\subset B$
(ii) $A-B=\varphi$
(iii) $A\cup B=B$
(iv) $A\cap B=A.$

Q. $A$ and $B$ are two events such that $P\left(A\right)=0.3,P\left(B\right)=0.8$ and $P(A\cap B)=0.16$. Find the probability that exactly one of the events happens.

Q.

If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8} and D = {7, 8, 9, 10}; find

(i) A ∪ B

(ii) A ∪ C

(iii) B ∪ C

(iv) B ∪ D

(v) A ∪ B ∪ C

(vi) A ∪ B ∪ D

(vii) B ∪ C ∪ D

Q. Whenever horses $a,b,c$ race together, their respective probabilities of winning the race are $0.3,0.5$ and $0.2$ respectively. If they race three times the probability that "the same horse wins all the three races" and the probability that $a,b,c$ each wins one race, are respectively (Assume no dead heat)

1. $\frac{10}{50};\frac{8}{50}$
2. $\frac{16}{100};\frac{3}{100}$
3. $\frac{8}{50};\frac{9}{50}$
4. $\frac{12}{50};\frac{15}{50}$

Q. Let $S=\{1,2,3,...,40\}$ and let A be a subset of such that no two elements in A have their sum divisible by 5. What is the maximum number of elements possible in A?

1. 10
2. 13
3. 17
4. 20

Q.

Let $F_1$ be the set of all parallelograms, $F_2$ the set of rectangles, $F_3$ the set of rhombuses, $F_4$ the set of squares and $F_5$ the set of trapeziums in a plane, then $F_1$ is equal to

1. 

2. 

3. 

4. 

Q.

For the three events A, B and C, P (exactly one of the events A or B occurs) = P(exactly one of the events B or C occurs) = P(exactly one of the events C or A occurs)=p and P(all the three events occur simultaneously)=$p^2$, where 0<p<1/2. Then the probability of at least one of the three events A, B and C occuring is

1. $\frac{3p+2p^2}{2}$
   

2. $\frac{p+3p^2}{4}$
   

3. $\frac{p+3p^2}{2}$
   

4. $\frac{3p+2p^2}{4}$

Q.

Let $F_1$ be the set of all parallelograms, $F_2$ the set of rectangles, $F_3$ the set of rhombuses, $F_4$ the set of squares and $F_5$ the set of trapeziums in a plane, then $F_1$ is equal to

1. 

2. 

3. 

4. 

Q. Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}. Verify that
(i) $\left(A\cup B\right)\text{'}=A\text{'}\cap B\text{'}$
(ii) $\left(A\cap B\right)\text{'}=A\text{'}\cup B\text{'}$.

Q.

IfS and T are two sets such that S has 21 elements, T has 32 elements, and

S∩T has 11 elements, how many elements does S ∪T have?

Q. Prove that if a is subset of phi then a = phi

$\mathit{Q}\mathbf{.}ProvethatifA\subset \varphi ,thenA=\varphi .$

Q. Any set $A$ such that $A\cup A=\{11,13,5,6\},$ then $A=$

1. $\{11,13,5,6\}$
2. $\{5,6\}$
3. $\left\{\right\}$

Q. write all subsets of {phi, 1}

Q.

IfA = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8} and D = {7, 8, 9, 10}; find

(i) A∪B

(ii) A∪C

(iii) B∪C

(iv) B ∪D

(v) A∪B ∪C

(vi) A∪B ∪D

(vii) B∪C ∪D

Q.

Write down all the subsets of the following sets:

(i) {a}

(ii) {a, b}

(iii) {1, 2, 3}

(iv) Φ