### Theory Of Sets Class 9th Mathematics Term 1 Tamilnadu Board Solution

##### Question 1.If A = {5, {5, 6}, 7}, which of the following is correct?A. {5, 6} ∈ AB. {5} ∈ AC. {7} ∈ AD. {6} ∈ AAnswer:In the given set we have been given that 2 elements and 1 setElements are 5, 7 and {5,6} is the setSo, {5,6} ∈ A.All the other options are not correctBecause {5} is a set not an element, so it does not belongs to A , but 5 ∈ AIn the same way {6}, {7} are sets not elements.Question 2.If X = {a, {b, c}, d}, which of the following is a subset of X?A. {a, b}B. {b, c}C. {c, d}D. {a, d}Answer:Given set X = {a, {b, c}, d}Here a, d is the elements {b, c} is the set present in the X.Subset:A is subset to B if all the elements in the A presents in the BE.g.,A = {9,8,4,6,7} B = {1,2,3,4,5,6,7,8,9,}Since all elements in A ⊆ B{a, b} is not subset because a is present in the set X but b is from other set and it is not an element in the set XIn the same way {b, c}, {c, d} are not subset of X.But a, d both are elements of set X, so {a, d} ⊆ XQuestion 3.Which of the following statements are true?i. For any set A, A is a proper subset of Aii. For any set A, Φ is a subset of Aiii. For any set A, A is a subset of AA. (i) and (ii)B. (ii) and (iii)C. (i) and (iii)D. (i), (ii) and (iii)Answer:i. For any set A, A is a proper subset of ANo set is proper subset of itself(or)Let us say A = {1,2,3,4}For a proper subset n(A) ≠ n(A)But here n(A) = n(A)So, it is not proper subset to itself.ii. For any set A, Φ is a subset of Ait is a property of null set, that it is subset to any other set except to itself (Φ)So, it is true.iii. For any set A, A is a subset of AIt is also property of subset that it is subset to itself and null set (Φ)So, it is also trueSince (ii) & (iii) is correct statement. option (B) is the correct answer.Question 4.If a finite set A has m elements, then the number of non-empty proper subsets of A isA. 2mB. 2m – 1C. 2m – 1D. 2(2m – 1 −1)Answer:Given Set A with ‘m’ number of elementsSo, the number of power subset for given set is given by 2m – 1∴ option (B) is true.Question 5.The number of subsets of the set {10, 11, 12} isA. 3B. 8C. 6D. 7Answer:Given set A {10, 11, 12}There are 3 elements in the set. ⇒ n(A) = 3The number of subset for a set is given by n(p(A)) = 2mWhere m is the no. of elements.So, no. of subsets = n(p(A)) = 23 = 8.∴ option(B) is the correct answer.Question 6.Which of the following is correct?A. {x:x2 = -1, x∈Z} = ΦB. Φ = 0C. Φ = {0}D. Φ = {Φ}Answer:B. Φ = 0 is false, because Φ is a null set, it is not equal to any element.C. Φ = {0} is false, because Φ should not contain any element in the set, but here it contains an element.D. Φ = {Φ} is also false, because null set will be considered as an element in the set.A. {x:x2 = -1, x∈Z} = ΦFor any value of x, x2 will not be going to -1. So, the set will be remained with no elements. So, this set will be considered as null set∴ option(A) is true.Question 7.Which of the following is incorrect?A. Every subset of a finite set in finiteB. P = {x : x – 8 = − 8} is a singleton setC. Every set has a proper subsetD. Every non – empty set has at least two subsetsAnswer:A. For a set to be subset to any set it should have less than or same number of elements. If a set contains finite no. of elements its subset also gets number finite in the set.So, this is true.B. The set is defined by P = {x : x – 8 = − 8}X -8 = -8X = 0∴ P = {0}This is a singleton set.This Is also true.D. It is a rule that non-empty set has 2 subsets they areThey are subset to itself and null set is the subset to that non-empty set.So, it is also true.C. It is a property that null set is proper set to every other set except to itself. This means Some sets only has proper subset.So, it is false.∴ option(C) is correct answer.Question 8.Which of the following is a correct statement?A. Φ ⊆ {a, b}B. Φ ∈ {a, b}C. {a} ∈ {a, b}D. a ⊆ {a, b}Answer:A. Φ ⊆ {a, b}We know that null set is subset to every other non-empty set. So, it is true.B. Φ ∈ {a, b}Null set consists of no elements, but here it contains elements. So, it is falseC. {a} ∈ {a, b}In the above {a} ⊂ {a, b} not {a} ∈ {a, b} since they are given as set but not element.D. a ⊆ {a, b}In the above {a} ∈ {a, b} not {a} ⊆ {a, b} since they are given as element but not as set.So, option(A) is the correct answer.Question 9.Which one of the following is a finite set?A. {x: x ∈ Z, x < 5}B. {x: x ∈ W, x ≥ 5}C. {x: x ∈ N, x > 10}D. {x: x is an even prime number}Answer:A. {x: x ∈ Z, x < 5}The above expression says that x defines the set. While the x belongs to integers which are less than 5.We know that Z contains -∞ to ∞But her it will contain elements from -∞ to 5So, this is an infinite set.B. {x: x ∈ W, x ≥ 5}The above expression says that x defines the set. While the x belongs to whole numbers(W) which is ≥ 5.We know that W contains 0 to ∞But her it will contain elements from 5 to ∞So, this is an infinite set.C. {x: x ∈ N, x > 10}The above expression says that x defines the set. While the x belongs to natural numbers(N) which greater than 10.We know that N contains 1 to ∞But her it will contain elements from 10 to ∞So, this is an infinite set.D. {x: x is an even prime number}The above expression says that x defines the set. While the x belongs to even prime number (prime numbers which are even). Fortunately, we have only one even prime number that is ‘2’So, this is finite series.Question 10.Given A = {5, 6, 7, 8}. Which one of the following is incorrect?A. Φ ⊆ AB. A ⊆ AC. {7, 8, 9} ⊆ AD. {5} ⊂ AAnswer:Given set A = {5, 6, 7, 8}A. Φ ⊆ AWe know that null set will be subset to every other non-empty set.So, it’s trueB. A ⊆ AWe know that a set is subset to itself. So, it’s true.C. {7, 8, 9} ⊆ AThe above set will not be a subset to A because set A didn’t contain element ‘9’. But it contains. So, it is false.D. {5} ⊂ AThe above set is power set to A because it has same element as in set A and the no. of elements in Set is less than set A. So, it is true.Question 11.If A = {3, 4, 5, 6} and B = {1, 2, 5, 6}, then A ∪ B =A. {1, 2, 3, 4, 5, 6}B. {1, 2, 3, 4, 6}C. {1, 2, 5, 6}D. {3, 4, 5, 6}Answer:A = {3, 4, 5, 6} & B = {1, 2, 5, 6}Here we are asked to find A ∪ BUnion means cub the elements of the both the given two sets.So,A ∪ B = {3, 4, 5, 6} ∪ {1, 2, 5, 6}= {1, 2, 3, 4, 5, 6}∴ option A is the correct answer.Question 12.The number of elements of the set {x: x ∈ Z, x2 = 1}isA. 3B. 2C. 1D. 0Answer:The set defined by {x: x ∈ Z, x2 = 1}This means the elements of the set is given by x2 = 1 and that x is belongs to integers(Z)x = 1 ⇒ x2 = 1x = -1 ⇒ x2 = (-1)2 = 1For the x = 1 & -1 the statement holds true.∴ 1, -1 are the elements of the setThe number of elements in the set is 2.Option B is the correct answer.Question 13.If n(X) = m, n(Y) = n and n(X ∩ Y) = p then n(X ∪ Y) =A. m + n + pB. m + n – pC. m – pD. m – n + pAnswer:Givenn(X) = m, n(Y) = n and n(X ∩ Y) = pn(X ∪ Y) = ?we know that n(A ∪ B) = n(A) + n(B) - n(A ∩ B)in the same way, n(X ∪ Y) = n(X) + n(Y) - n(X ∩ Y)n(X ∪ Y) = m + n – p∴ option B is the correct answer.Question 14.If U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {2, 5, 6, 9, 10} then A’ isA. {2, 5, 6, 9, 10}B. ΦC. {1, 3, 5, 10}D. {1, 3, 4, 7, 8}Answer:Given,U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {2, 5, 6, 9, 10}We have to find A’ (A compliment)A’(A compliment) means we have to find the eliminate the elements of sets A in U.∴ A’ = {1, 3, 4, 7, 8}Option D is the correct answer.Question 15.If A B, then A – B isA. BB. AC. ΦD. B – AAnswer:Given that A B that means B contains all the elements that A contains.e.g., A = {9, 8, 4, 6} B = {9, 8, 4, 6, 7, 0, 1}we can see that B contains all the elements of BSo, A BLet us consider two sets which is A BA = {9, 8, 4, 6} B = {9, 8, 4, 6, 7, 0, 1}A – B means we eliminate all the elements of B from A.A – B = {9, 8, 4, 6} - {9, 8, 4, 6, 7, 0, 1}= {} ⇒ null set.Option D is the correct answer.Question 16.If A is a proper subset of B, then A ∩ B =A. AB. BC. ΦD. A ∪ BAnswer:Given, A is a proper subset of BLet us consider two set as A is a proper subset of BA = {9, 8, 7, 6, 5, 4} B = {1, 2, 3, 4, 5, 6, 7, 8, 9}A ∩ B means we have write all the common elements of both the sets.A ∩ B = {9, 8, 7, 6, 5, 4} ∩ {1, 2, 3, 4, 5, 6, 7, 8, 9}= {4, 5, 6, 7, 8, 9}The resultant set is same as AOption A is the correct answer.Question 17.If A is a proper subset of B, then A ∪ BA. AB. ΦC. BD. A ∩ BAnswer:Given, A is a proper subset of BLet us consider two set as A is a proper subset of BA = {9, 8, 7, 6, 5, 4} B = {1, 2, 3, 4, 5, 6, 7, 8, 9}A ∪ B means we must combinedly write all the elements of both the sets.A ∪ B = {9, 8, 7, 6, 5, 4} ∪ {1, 2, 3, 4, 5, 6, 7, 8, 9}= {1, 2, 3, 4, 5, 6, 7, 8, 9}The resultant set is same as BOption C is the correct answer.Question 18.The shaded region in the adjoint diagram represents A. A – BB. A’C. B’D. B – AAnswer:The venn diagram contains 2 sets, and they are overlapped which means they have common elements in it.In the venn diagram B is completely shaded Except the overlapped part with A that means “it shows the elements of B by eliminating the common elements of A”Since we are eliminating the elements of A in B this can be written as B-A.Option D is the correct answer.Question 19.If A = {a, b, c}, B = {e, f, g}, then A ∩ B =A. ΦB. AC. BD. A ∪ BAnswer:Given, A = {a, b, c}, B = {e, f, g}We have to find A ∩ BA ∩ B = {a, b, c} ∩ {e, f, g} = ΦSince there is no common elements in the both the sets the resultant is null set.Option A is the correct answer.Question 20.The shaded region in the adjoining diagram represents A. A – BB. B – AC. A Δ BD. A’Answer:The green shaded represents A-B and the blue shaded part represents B-AThe overall shaded region will the union of A-B & B-A∴ (A-B) ∪ (B-A) = A Δ B⇒ The shaded region represents A Δ B of the venn diagram.

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