Sets EXERCISE 1(b)

EXERCISE 1 (B)

1. Which of the following are examples of the null set?

(I) Set of odd natural number divisible by 2.

Ans: Null set 

(II) Set of even prime numbers.

Ans: Not a Null set

(III) {x: x is a natural number, x<5 and simultaneously x>7}

Ans: Null set 

(IV) {y: y is a point common to any two parallel lines}.

Ans: Null sel

2. Which of the following sets are empty sets?

(I) A={x: x²-5=0 and x is rational}

Ans: Empty 

(II) B={x : x²=26, and x is an odd integer}

Ans: Empty 

(III) Set of all even natural numbers divisible by 10

Ans: Not empty 

(IV) {x: x is a natural number, x<6 and simultaneously x>18}

Ans: Empty 

3. Which of the following sets are finite or infinite?

(I) The set of months of a year

Ans: Finite 

(II) {1,2,3,.......}

Ans: Infinite 

(III) {1,2,3,.....,......,99,100}

Ans: Finite 

(IV) The Set of positive integers greater than 100

Ans: Infinite 

(V) The set of prime numbers less than 99.

Ans: Finite

4. Which of the following pairs of Sets are equal? Justify your answer

(I) A={x: x is a letter in the word "LOYAL"}

 B={x : x is a letter of the word "ALLOY"}

Ans: A=B

(II) A={x: x ∈ Z and x² ≤ 8}

 B={x: x ∈ R and x²-4x+3=0}

Ans: A≠B

(III) A={n: n∈ Z and n²≤ 4}

B={x: x ∈ R and x²-3x+2=0}

Ans: A≠B

5. Which of the following sets are equal? Give reasons Ø,{0}, {Ø}, 0

Ans: No two of these sets are equal

6. Which of the following sets are equivalent?

    Ø, {0} , {Ø}

Ans:  {0} and {Ø} are equivalent 

7. Is the set A={x : x^{3=8 and 2x+3=0} empty? Justify.}

^{Ans: Empty} 

^{Art-4. Sub-Set, Super-Set}

^{If every member of a set A is a member of a set B,} 

^{Then A is called sub-set of B and B is called super-set} 

^{of A.}

^{Or if x∈ A =>  x∈B, then A is a sub set of B and B is a} 

^{Super set of A .}

^{We write these as A ⊆ B and B⊇ A.}

^{Thus A⊆B means A is contained in B or B contains A.}

^{If A is not a subset of B , we write it as A⊊B}

^{Note 1. Since every element of A belongs A} 

^{∴ A⊆A   => every set is sub set of itself .}

^{2. The empty set Ø is taken as a sub-set of every set} 

^{Example: Let A={1,2,3,4,5,6,8,10}}

^{and B={2,4,6,10} , C={1,2,7,8}, D={2,7,8,1}}

^{Now every element of B is an element of A} 

^{∴ B⊆A}

^{Again 7∈ C , but 7∉A}

^{∴ C⊊A i.e. C is not a sub set of A.}

^{Now every member of D is a member of} C and every 

Member of C is a member of D.

∴     C⊊D and D⊆C

^{Equality of Sets}

^{Two Sets A and B are said to be equal if both have}

^{the same elements . In order words, two sets A and B}

are equal when every element of A is an element of B 

And every element of B is element of A.

I.e. If A⊆B and B⊆A, then A=B.

Example.   A={1,2,3,4,5,6,7,8,9,10}

                    B={x: x is a natural number and 1≤ x ≤10

Here A=B

Note : Difference between equal and equivalent sets

In equal sets, number of elements of both sets is same and also elements of both sets are same . In equivalent sets, number of elements of both sets is same but element may be different.

Proper sub-set 

A set A is said to be a proper sub-set of B if A⊆B and A≠B

Note 1. A is proper sub-set of B is denoted in A ⊂