10 sets example with answers
Answers
Answer:
. Write the solution set of the equation x2 – 4=0 in roster form.
Solution: x2 – 4 = x2 – 22 = (x-2) (x+2)
x=2,-2
Thus, A = {-2,2}
2. Write the set A = {1, 4, 9, 16, 25, . . . } in set-builder form.
Solution: If we see the pattern here, the numbers are squares of natural numbers, such as:
12 = 1
22 = 4
32 = 9
42 = 16
And so on.
A = {x : x is the square of a natural number}
Or we can write;
A = {x : x = n2 , where n ∈ N}
Empty Set: A set with no elements. Also called a void set or null set
Finite & Infinite Set: A set with a definite number of elements (even with zero elements) is a finite set otherwise the set is infinite
Equal sets: Two sets if having the same elements is called equal sets.
3. Write an example of finite and infinite set in set builder form.
Solution:
Finite set, A = {x : x ∈ N and (x – 1) (x –2) = 0}
Infinite Set, B = {x : x ∈ N and x is prime}
4. Write an example of equal sets.
Solution: Let there be two sets A and B
A is the set of letter in “ALLOY”
B is the set of letter in “LOYAL”
Hence,
A = {A,L,L,O,Y}
B = {L,O,Y,A,L}
Therefore, in both the sets the elements are the same.
So, A = B.
Subsets: A set A is said to be a subset of set B if every element of A is also an element of B
Symbolically, A ⊂ B if a ∈ A ⇒ a ∈ B.
Power Set: Collection of all subsets of a set.
5. Write the subsets of {1,2,3}.
Solution: Let A = {1,2,3}
The subsets of A are: φ, {1},{2},{3},{1,2},{2,3},{1,3},{1,2,3}
6. Write {x: x ∈ R, 3 ≤ x ≤ 4} as intervals.
Solution: {x: x ∈ R, 3 ≤ x ≤ 4} = [3, 4]
7. Write {6, 12} in set builder form.
Solution: Let A = {6, 12}
If we write in set builder form, then;
A={x: x ∈ R, 6 < x ≤ 12}
8. If set A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}. Then write the universal set for all three sets.
Solution: If U is the universal set for sets A, B and C, then:
U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
Recheck:
A ⊂ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
B ⊂ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
C ⊂ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
Also, read: Sets Subset And Superset
Union of sets: If A and B are two sets, then A union B will have all the elements of set A and set B. It is represented as A ∪ B.
Intersection of sets: If A and B are two sets, then A intersection B will have common elements of set A and set B. It is represented as A ∩ B.
9. If A = { 2, 4, 6, 8} and B = { 6, 8, 10, 12}. Find A ∪ B.
Solution: A ∪ B = { 2, 4, 6, 8, 10, 12}
10. If A = { 2, 4, 6, 8} and B = { 6, 8, 10, 12}. Find A ∩ B.
Solution: A ∩ B = { 6, 8 }
Also, see: Operation On Sets Intersection Of Sets And Difference Of Two Sets
Properties of Union of Sets
Properties of Intersection of sets
A ∪ B = B ∪ A (Commutative law)
A ∩ B = B ∩ A (Commutative law)
A ∪ B ) ∪ C = A ∪ ( B ∪ C) (Associative law )
( A ∩ B ) ∩ C = A ∩ ( B ∩ C ) (Associative law)
A ∪ φ = A (Law of identity element, φ is the identity of ∪)
φ ∩ A = φ, U ∩ A = A (Law of φ and U)
A ∪ A = A (Idempotent law)
A ∩ A = A (Idempotent law)
U ∪ A = U (Law of U)
A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C ) (Distributive law )
11. If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8}. Find A ∪ B ∪ C.
Solution: A ∪ B ∪ C = {1, 2, 3, 4, 5, 6, 7, 8}
12. If A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15}. Find A ∩ (B ∪ C).
Solution: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
= {7, 9, 11} ∪ {11}
= {7, 9, 11}
Difference of sets: If A and B are two sets, then the difference of set A and set B is a set which has elements of only set A not B. It is represented as A – B.
13. If A = { 1, 2, 3, 4, 5, 6}, B = { 2, 4, 6, 8 }. Find A – B and B – A.
Solution: A – B = { 1, 3, 5 }
B – A = { 8 }
Clearly, A – B ≠ B – A
Complement of set: If A is a subset of universal set U, then complement of a set A is the set which does not have any elements of A. It is denoted as A’.
A′ = {x : x ∈ U and x ∉ A }
A′ = U – A
14. If U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {1, 3, 5, 7, 9}. Find A′.
Solution: A′ = { 2, 4, 6, 8,10 }