De Morgan's Laws

Q. If A, B and C are any three sets, then $A-(B\cup C)$ is equal to

1. $(A-B)\cup (A-C)$
2. $(A-B)\cap (A-C)$
3. $(A-B)\cup C$
4. $(A-B)\cap C$

Q. Let $A,B$ are two non-empty sets and $U$ be the universal set. Then which of the following statements is/are true?
$1.n(A\cup B{)}^{\prime }=n(A^{\prime }\cap B^{\prime })$
$2.$ If $A\cap B=\varphi ,$ then $A^{\prime }\cup B^{\prime }=U$
$3.$ If $A\cup B=U,$ then $A^{\prime }\cap B^{\prime }=\varphi$
$4.$ If $A\subset B,$ then $A^{\prime }\cup B^{\prime }=(A\cap B{)}^{\prime }$
(where $\varphi$ denote the null set)

1. $1,2$ and $3$ only
2. $1$ and $3$ only
3. $1,2$ and $4$ only
4. All statements are true

Q. The no. of elements in $(A\cup B{)}^{\prime }$ is the same as the number of elements in $A^{\prime }\cap B^{\prime }.$

1. False
2. True

Q. In an examination, out of $100$ students, $75$ passed in English, $60$ passed in Mathematics and $45$ passed in both English and Mathematics. The number of students who passed in none of the two subjects, is
(Assume all students gave both the exams)

1. $5$
2. $10$
3. $20$
4. $40$

Q. Let \(n(U) = 700 , n(A) = 200 , n(B) = 300,
n(A∩B)=100, then n(A' ∩ B')=

1. $400$
2. $600$
3. $500$
4. $300$

Q. If $A^{\prime }\subset B,$ then $(A\cap B^{\prime }{)}^{\prime }=$

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

Q. Let \(n(U) = 700 , n(A) = 200 , n(B) = 300,
n(A∩B)=100, then n(A' ∩ B')=

1. $400$
2. $600$
3. $500$
4. $300$

Q. For two sets $A$ and $B$, $(A\cup B)\cap (A^{\prime }\cup B^{\prime })=$

1. $A\cap B$
2. $A\cup B$
3. $A-B$
4. $A\Delta B$

Q. Let $A$ and $B$ are two sets. Then $n(A^C\cap B^C)=$

1. $n\left(U\right)–n(A\cap B)$
2. $n\left(U\right)–n(A\cup B)$
3. $n\left(U\right)–n(A^C\cup B)$
4. $n\left(A\right)+n\left(B\right)$

Q. For any two sets $A$ and $B,$ if $(A\cup B{)}^{\prime }=A^{\prime }\cup B^{\prime },$ then

1. $A\subseteq B$
2. $B\subseteq A$
3. $A^{\prime }=B$
4. $A=B$

Q. The number of elements in $(A\cap B{)}^{\prime }$ is the same as the number of elements in $A^{\prime }\cup B^{\prime }.$

1. False
2. True

Q. For two sets $A$ and $B$, $(A\cup B)\cap (A^{\prime }\cup B^{\prime })=$

1. $A\cap B$
2. $A\cup B$
3. $A-B$
4. $A\Delta B$

Q.

If the number of elements in each of universal set, set A and set B are represented as $n\left(U\right)$, $n\left(A\right)$, $n\left(B\right)$ respectively, then the number of elements that are not in set A and not in set B is given by ___.

1. $n\left(AU{B}^{\prime }\right)$

2. $n(A\cap B{)}^{\prime }$

3. $n\left(U\right)-n(A\cap B)$

4. $n\left(U\right)-n(A\cup B)$

Q.

If the number of elements in each of universal set, set A and set B are represented as $n\left(U\right)$, $n\left(A\right)$, $n\left(B\right)$ respectively, then the number of elements that are not in set A and not in set B is given by ___.

1. $n\left(AU{B}^{\prime }\right)$

2. $n(A\cap B{)}^{\prime }$

3. $n\left(U\right)-n(A\cap B)$

4. $n\left(U\right)-n(A\cup B)$

Q. Let $n\left(U\right)=700$, $n\left(A\right)=200$, $n\left(B\right)=300$ and $n(A\cap B)=100$, Then $n(A^C\cap B^C)=$

1. $100$
2. $300$
3. $500$
4. $450$

Q.

In a survey of 100 students, the number of students studying the various languages were found to be: English only 18, English but not Hindi 23, English and Sanskrit 8, English 26, Sanskrit 48, Sanskrit and Hindi 8 , no language 24. Find:

(ii) How many students were studying English and Hindi?

Q.

$(A-B)-(B-C)=$___

1. A - B

2. B - C

3. A - C

4. A - B - C

Q. If $A^{\prime }\subset B,$ then $(A\cap B^{\prime }{)}^{\prime }=$

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

Q. If $(A^{\prime }\cap B^{\prime }{)}^{\prime }=\varnothing ,$ then $(A\cup B{)}^{\prime }=$

1. $U$
2. $\varnothing$
3. $A\cup B$
4. $A$