Set-Theory Practise Question

It is De-Morgan’s Law

In set theory, De Morgan's Laws describe the relationship between the complement of unions and intersections of sets. They are similar to the laws in Boolean algebra but applied to set operations.

The two main laws in set theory are:

Hence, the correct answer is option c. De-Morgan’s Law

Given,

A={1,2,3,4} and B={3,4,5}

For AΔB we know that,

AΔB=(A∪B)−(A∩B)---------(i)

(AuB) = {1,2,3,4}  u {3,4,5} ={ 1,2,3,4,5}

(AnB) = {1,2,3,4}  n {3,4,5} ={3,4}

then equation(i) becomes

AΔB=(A∪B)−(A∩B)

={1,2,3,4,5} - {3,4}

={1,2,5}

Cardinality of set: It is the number of distinct elements in any finite set denoted by n(set)

so, the cardinality of AΔB

he set $\}$ has 3 elements

The cardinality of the power set of $n$

Since AΔB has 3 elements, its power set will have:2^3=8

Hence, the correct answer is option d.8

A well-defined set is one in which it is possible to determine whether any given object is an element of the set. The term "intelligent students" is subjective and can vary from person to person, hence it is not a well-defined set.

Hence, the correct answer is option b. not a well defined set

Given,

A={1,2,3,4}

Total number of subsets of a set(n)=4 i.e Set A Containing 4 elements so n=4

The total number of subsets is 2^4= 16 which also include empty set

but according to the questions, we have to find the number of non-empty subset

So, For that we have

total number of subsets -1

$16-1=15$.

Hence, the correct answer is option d. 15

Given,  A={ɸ,{ɸ}} i.e total number of element is 2

n=2

Using power set formula = 2^n

so, Power of set = 2^n = 2^2  = 4

The sets are {∅,{∅},{{∅}},{∅,{∅}}}

Hence, the correct answer is option d. {∅,{∅},{{∅}},{∅,{∅}}}

Given,

Set of order(n)= 5

The number of subsets is 2^n = 2^5 = 32

Hence, the correct answer is option b. 32

Given,

A=(-3,2) i.e {-2,-1,0,1}

and B=(-2,3) i.e {-1, 0,1,2}

AnB= {-2,-1,0,1} n {-1, 0,1,2} = {-1,0,1} i.e (-2,2)

Hence, the correct answer is option c. (-2,2)

Here, in the questions it asked to find the set of all prime numbers; which is not defined that's means it is infinite

So, It is an infinite set

Hence, the correct answer is option c. an infinite set

$Given,$

f(x)=(x−2) / (2−x) , x≠2x 

then,

=(x−2) / -(x−2)

= -1

Hence, the correct answer is option d. -1

From the idempotent laws

AuA=A       & AnA=A

Hence, the correct answer is option d. AuA=A

Given,

A=[-3,2] i.e {-3,-2,-1,0,1,2}

and B=(-2,3) i.e {-1,0,1,2}

AuB = {-3,-2,-1,0,1,2} u {-1,0,1,2} ={-3,-2,-1,0,1,2} = [-3,2]

Hence, the correct answer is option b. [-3,2]

For calculating proper subsets we can use  2^𝑛 − 1 formula

Hence, the correct answer is option b. 2^n - 1

[a,b] for calculating this first of all we need to know that [...] represents belong to that means the starting and ending element must lies in the set

so from the questions, [a,b] = {x: a ≤ x ≤ b }

Hence, the correct answer is option a. {x: a ≤ x ≤ b }

(- ∞, ∞)for calculating this first of all we need to know that (...) represents does't belong to set that means the starting and ending element doesn't lies in the set

so from the questions, . (- ∞, ∞) = {x: - ∞ < x < ∞}

Hence, the correct answer is option b.. {x: - ∞ < x < ∞}

We know that, the modulus represents -ve & +ve sign

so, from the questions,

|x|=1

∴ x={-1,1}

Hence, the correct answer is options c. {-1,1}