Practice set 1.3 Sets Class 9th Mathematics Part I MHB Solution
Practice Set 1.3
Question 1.
If A = {a, b, c, d, e}, B = {c, d, e, f}, C = {b, d} and D = {a, e}.
then which of the following statements are true and which are false?
i. C ⊆ B ii. A ⊆ D
iii. D ⊆ B iv. D ⊆ A
v. B ⊆ A vi. C ⊆ A
Answer:
We consider the following Venn diagram to check whether the above statements are true or not.
Now, we have A = {a, b, c, d, e}, B = {c, d, e, f}, C = {b, d} and D = {a, e}.
So, it is clear from above that every element of set C and set D is also an element of set A.
∴ C ⊆ A and D ⊆ A
Hence, (iv), (vi) are true and (ii) is false.
Also, it is clear from above that every element of set B is not an element of set A.
∴ (v) is false.
Similarly, it is clear that every element of set C is not an element of set B.
∴ (i) is false.
Also, every element of set D is not an element of set B.
∴ (iii) is false.
Question 2.
Take the set of natural numbers from 1 to 20 as universal set and show set X and Y using Venn diagram.
i. X = {x | x ∈ ℕ, and 7 < x < 15}
ii. Y = {y | y ∈ ℕ, y is prime number from 1 to 20}
Answer:
Consider the universal set:
U = {u | u ∈ ℕ, 1 ≤ x ≤ 20}
= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}
X = {x | x ∈ ℕ, 7 < x < 15}
= {8, 9, 10, 11, 12, 13, 14}
Y = {y | y ∈ ℕ, y is prime number from 1 to 20}
= {2, 3, 5, 7, 11, 13, 17, 19}
Question 3.
U = {1, 2, 3, 7, 8, 9, 10, 11, 12}
P = {1, 3, 7, 10}
then (i) show the sets U, P and P � by Venn diagram. (ii) Verify (P’)’ = P
Answer:
U = {1, 2, 3, 7, 8, 9, 10, 11, 12}
P = {1, 3, 7, 10}
P’ = {x | x ∈ U and x ∉ P} = {2, 8, 9, 11, 12}
(i) The Venn diagram showing U, P and P’ is:
(ii) Now, we consider (P’)’
(P’)’ = {x | x ∈ U and x ∉ P’}
= {1, 3, 7, 10}
= P
Hence, (P’)’ = P.
Question 4.
A = {1, 3, 2, 7} then write any three subsets of A.
Answer:
We know that any set S such that every element of S is an element of the set A is said to be a subset of A.
∴ Consider B = {1, 2, 3}, C = {1, 3} and D = {2, 3, 7}
It is clear from above that every element of the sets B, C and D is an element of set A, so, B, C and D are subsets of A.
Also, note that
Every set is a subset of itself, i. e. A ⊆ A and
Empty set is a subset of every set, i. e. ϕ ⊆ A.
So, A and ϕ are also subsets of A.
Question 5.
(i) Write the subset relation between the sets.
P is the set of all residents in Pune.
M is the set of all residents in Madhya Pradesh.
I is the set of all residents in Indore.
B is the set of all residents in India.
H is the set of all residents in Maharashtra.
(ii) Which set can be the universal set for above sets?
Answer:
(i) Consider the following Venn diagram for the above sets:
(ii) From the above Venn diagram, it is clear that set B can serve as the universal set for above sets.
Question 6.
Which set of numbers could be the universal set for the sets given below?
i. A = set of multiples of 5,
B = set of multiples of 7.
C = set of multiples of 12
Answer:
A universal set is the set which can accommodate all other sets in it, i.e. every element of all the sets should be an element of the universal set.
A = set of multiples of 5,
B = set of multiples of 7.
C = set of multiples of 12
Consider the set of all integers, I = {………, -2, -1, 0, 1, 2,………}
We see that every element of the sets A, B and C is an element of the set I.
So, set I can serve as the universal set for the sets A, B and C.
Question 7.
P = set of integers which multiples of are 4.
T = set of all even square numbers.
Answer:
A universal set is the set which can accommodate all other sets in it, i.e. every element of all the sets should be an element of the universal set.
P = set of integers which are multiples of 4
= {x | x ∈ ℤ and x is a multiple of 4}
T = {y | y is an even square number}
Consider the set of all integers, I = {………, -2, -1, 0, 1, 2,………}
We see that every element of both the sets P and T is an element of the set I.
So, set I can serve as the universal set for the sets P and T.
Question 8.
Let all the students of a class is a Universal set. Let set A be the students who secure 50% or more marks in Maths. Then write the complement of set A.
Answer:
Let U be the universal set.
Then, U = Set of all students of a class
A = Set of students who secure 50% or more marks in Maths
We know that a set A’ is said to be the complement of the set A if it contains all the elements of the universal set U, which are not in the set A.
∴The complement of set A is:
A’ = Set of all students who do not secure 50% or more marks in Maths
Or A’ = Set of all students who secured less than 50% marks in Maths
∴ A'= U - A
