De Morgan's Law

Q.

Find the number of solutions for $4\left\{x\right\}=x+\left[x\right]$, where $\{.\}$, [.] represents fractional part and greatest integer function.

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Q. Fill in the blanks to make each of the following a true statement: (i) (ii) Φ′ ∩ A = … (iii) (iv)

Q.

Let ∪ = {x: x is a natural number and x ≤ 10}

A = {a: a is an even natural number and a < 10}

B = {b: b is a prime number and b ≤ 9}

Then A^I ∩ B^I is

1. {1, 9, 10}

2. {2, 3, 5, 7}

3. { 3, 5, 7}

4. {2, 4, 6, 8}

Q. Out of all the patients in a hospital $89\%$ are found to be suffering from heart ailment and $98\%$ are suffering from lungs infection. If $K\%$ of them are suffering from both ailments, then $K$ can not belong to the set:

1. $\{84,87,90,93\}$
2. $\{84,86,88,90\}$
3. $\{79,81,83,85\}$
4. $\{80,83,86,89\}$

Q.

In a class of 60 students, 30 opted for NCC, 32 opted for NSS and 24 opted for both NCC and NSS. If one of these students is selected at random, find the probability that
(i) The student opted for NCC or NSS.
(ii) The student has opted neither NCC nor NSS.
(iii) The student has opted NSS but not NCC.

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) $A^{\prime }$ (ii) $B^{\prime }$ (iii) ${(A\cup C)}^{\prime }$

(iv) ${(A\cup B)}^{\prime }$ (v) $(A{)}^{\prime }$ (vi) ${(B-C)}^{\prime }$

Q.

A = {1, 2, 3}, B = {2, 3, 4}, C = {4, 5}. Find (A ∩ B) $\times$ C.

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

1. A
2. B
3. A - B
4. B - A

Q. If A × B ⊆ C × D and A × B ≠ ϕ, prove that A ⊆ C and B ⊆ D.

Q. If $X$ and $Y$ are two sets and $A\text{'}$ denotes the complement of $A$, then $X\cap (X\cup Y)\text{'}$ is equal to

1. X
2. Y
3. $\varphi$
4. $X\cap Y$

Q.

Examine whether the following statements are true or false:

(i) {a, b} ⊄ {b, c, a}

(ii) {a, e} ⊂ {x: x is a vowel in the English alphabet}

(iii) {1, 2, 3} ⊂{1, 3, 5}

(iv) {a} ⊂ {a. b, c}

(v) {a} ∈ (a, b, c)

(vi) {x: x is an even natural number less than 6} ⊂ {x: x is a natural number which divides 36}