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 : x3=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 ⊂
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