Mathematics
Indicate whether the given statement is true or false :
(i) {Triangles} ⊆ {Quadrilaterals}
(ii) {Squares} ⊆ {Rectangles}
(iii) {Rhombuses} ⊆ {Parallelograms}
(iv) {Natural numbers} ⊆ {Whole numbers}
(v) {Integers} ⊆ {Whole numbers}
(vi) {Composite numbers} ⊆ {Odd numbers}
Sets
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Answer
(i) False
Reason — A triangle is a polygon with 3 sides, whereas a quadrilateral is a polygon with 4 sides. Since no triangle can be a quadrilateral, the set of triangles is not a subset of the set of quadrilaterals.
(ii) True
Reason — A square is defined as a special type of rectangle where all four sides are equal. Since every square satisfies the properties of a rectangle, the set of squares is a subset of the set of rectangles.
(iii) True
Reason — A rhombus is a quadrilateral with both pairs of opposite sides parallel and all sides equal. Since it satisfies the definition of a parallelogram (a quadrilateral with two pairs of parallel sides), the set of rhombuses is a subset of the set of parallelograms.
(iv) True
Reason — Natural numbers (N) are {1, 2, 3, …} and whole numbers (W) are {0, 1, 2, 3, ….}. Since every natural number is also a whole number, {Natural numbers} ⊆ {Whole numbers}.
(v) False
Reason — Integers include both, negative and positive numbers (…., -2, -1, 0, 1, 2, ….), whereas whole numbers consist only of zero and positive counting numbers. Since negative integers are not whole numbers, the set of integers is not a subset of whole numbers.
(vi) False
Reason — Composite numbers are numbers with more than two factors, such as 4, 6, 8, 9, 10, … . Many composite numbers (like 4, 6, and 8) are even, so the set of composite numbers is not a subset of the set of odd numbers.
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Related Questions
For each of the following pairs of sets, identify the disjoint and overlapping sets :
(i)
A = {x : x is a prime number, x < 8}.
B = {x : x is an even natural number, x < 8}.(ii) C = {x : x ∈ N, x < 10} and D = {x : x ∈ N, x is a multiple of 5}.
(iii) E = {x : x = 4n, n ∈ N} and F = {x : x = 9n, n ∈ N}.
(iv) G = {x : x = 8n, n ∈ N and n < 7} and H = {x : x = 9n, n ∈ N and n < 7}.
State in each case, whether the given statement is true or false :
(i) If A is the set of all non-negative integers, then 0 ∈ A.
(ii) If B is the set of all consonants, then c ∈ B.
(iii) If C is the set of all prime numbers less than 80, then 57 ∈ C.
(iv) {x : x ∈ W, x + 5 = 5} is a singleton set.
(v) If D = {x : x ∈ W, x < 4}, then n(D) = 4.
(vi) {a, b, c, 1, 2, 3} is not a set.
(vii) {1, 2, 3, 1, 2, 3, 1, 2, 3,……………} is an infinite set.
(viii) 0 ∈ Φ.
(ix) {3, 5} ∈ (1, 3, 5, 7, 9).
Write down all possible subsets of each of the sets given below :
(i) {1}
(ii) {3, 4}
(iii) {2, 3, 5}
(iv) Φ
(v) {c, d, e}
(vi) {a, b, c, d}
Write down all possible proper subsets of each of the sets given below :
(i) {x}
(ii) {p, q}
(iii) {m, n, p}
(iv) {1, 2, 3, 4}