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Ex 1.5, 3 - Taking set of natural numbers as universal set - Venn Diagrams - Compelment

 


  1. Chapter 1 Class 11 Sets
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Ex1.5,3 (Method 1) Taking the set of natural numbers as the universal set, write down the complements of the following sets: (i) {x: x is an even natural number} Universal set is the set of natural numbers {x: x is an even natural number}’ = {x: x is an odd natural number} Ex1.5,3(Method 2) Taking the set of natural numbers as the universal set, write down the complements of the following sets: (i) {x: x is an even natural number} Universal set is the set of natural numbers So, U = {1, 2, 3, 4, 5…..} {x: x is an even natural number} = {2, 4, 6, 8, 10,….} {x: x is an even natural number}’ = {2, 4, 6, 8, 10,….}’ = U – {2, 4, 6, 8, 10,….} = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10,….} – {2, 4, 6, 8, 10,…} = {1, 3, 5, 7, 9, 11, 13, …..} = {x: x is an odd natural number} Ex1.5,3 (Method 1) Taking the set of natural numbers as the universal set, write down the complements of the following sets: (ii) {x: x is an odd natural number} Universal set is the set of natural numbers {x: x is an odd natural number}’ = {x: x is an even natural number} Ex1.5,3 (Method 2) Taking the set of natural numbers as the universal set, write down the complements of the following sets: (ii) {x: x is an odd natural number} Universal set is the set of natural numbers So, U = {1, 2, 3, 4, 5…..} {x: x is an odd natural number} = {1, 3, 5, 7, 9, 11, 13, …..} {x: x is an odd natural number}’ = {1, 3, 5, 7, 9, 11, 13, …..}’ = U – {1, 3, 5, 7, 9, 11, 13, …..} = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10,….} – {1, 3, 5, 7, 9, 11, 13, …..} = {2, 4, 6, 8, 10,….} = {x: x is an even natural number} Ex 1.5,3(Method 1) Taking the set of natural numbers as the universal set, write down the complements of the following sets: (iii) {x: x is a positive multiple of 3} U = {1,2,3,4,5…..} {x: x is a positive multiple of 3}´ = {x: x ∈ N and x is not a multiple of 3} Ex 1.5,3(Method 2) Taking the set of natural numbers as the universal set, write down the complements of the following sets: (iii) {x: x is a positive multiple of 3} Universal set is the set of natural numbers So, U = {1, 2, 3, 4, 5…..} {x: x is a positive multiple of 3} = {1 × 3 , 2 × 3 , 3 × 3 , 4 × 3 , ….} = {3 , 6 , 9, 12, ….} {x: x is a positive multiple of 3}´ = {3 , 6 , 9, 12, ….}’ = U – {3 , 6 , 9, 12, ….} = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …..} – {3 , 6 , 9, 12, ….} = {1, 2, 4, 5, 7, 10, 11, 13,…..} = {x: x ∈ N and x is not a multiple of 3} Ex1.5,3 (Method 1) Taking the set of natural numbers as the universal set, write down the complements of the following sets: (iv) {x: x is a prime number} U is set of natural numbers {x: x is a prime number}´ = {x: x ∈ N and x is a composite number and x = 1} Ex1.5,3 (Method 2) Taking the set of natural numbers as the universal set, write down the complements of the following sets: (iv) {x: x is a prime number} U = {1,2,3,4,5…..} {x: x is a prime number} = {2,3,5,7,11,13…} {x: x is a prime number}´ = {2,3,5,7,11,13…}’ = U – {2,3,5,7,11,13…} = {1, 2, 3, 4, 5, 6, ….} – {2,3,5,7,11,13…} = {1,4,6,8,9,10,12,14,….} = {x: x ∈ N and x is not a prime number and x = 1} = {x: x ∈ N and x is a composite number and x = 1} Ex1.5,3 Taking the set of natural numbers as the universal set, write down the complements of the following sets: (v) {x: x is a natural number divisible by 3 and 5} U = {1,2,3,4,5…..} {x: x is a natural number divisible by 3 and 5}’ = {x: x is a natural number that is not divisible by 3 or 5} Ex1.5,3(Method 1) Taking the set of natural numbers as the universal set, write down the complements of the following sets: (vi) {x: x is a perfect square} U = {1,2,3,4,5…..} {x: x is a perfect square}´ = {x: x ∈ N and x is not a perfect square} Ex1.5,3(Method 2) Taking the set of natural numbers as the universal set, write down the complements of the following sets: (vi) {x: x is a perfect square} Universal set is the set of natural numbers So, U = {1, 2, 3, 4, 5…..} {x: x is a perfect square} = {12 , 22 , 32 , 42 , ….} = {1, 4, 9 , 16, ….} {x: x is a perfect square}´ = {1, 4, 9 , 16, ….}’ = U – {1, 4, 9 , 16, ….} = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …..} – {1, 4, 9 , 16, ….} = {2, 3 , 5, 6, 7, 10, 11, …..} = {x: x ∈ N and x is not a perfect square} Ex1.5,3 (Method 1) Taking the set of natural numbers as the universal set, write down the complements of the following sets: (vii) {x: x is perfect cube} Universal set is the set of natural numbers {x: x is a perfect cube}´ = {x: x ∈ N and x is not a perfect cube} Ex1.5,3 (Method 2) Taking the set of natural numbers as the universal set, write down the complements of the following sets: (vii) {x: x is perfect cube} Universal set is the set of natural numbers So, U = {1, 2, 3, 4, 5…..} {x: x is a perfect cube} = {13 , 23 , 33 , 43 , ….} = {1, 8, 27 , 64, ….} {x: x is a perfect cube}´ = {1, 8, 27 , 64, ….}’ = U – {1, 8, 27 , 64, ….} = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …..} – {1, 8, 27 , 64, ….} = {2, 3 , 4, 5, 6, 7, 9, 10, 11, …..} = {x: x ∈ N and x is not a perfect cube} Ex1.5,3 Taking the set of natural numbers as the universal set, write down the complements of the following sets: (viii) {x: x + 5 = 8} U = {1,2,3,4,5…..} x + 5 = 8 x = 8 – 5 = 3 {x: x + 5 = 8}´ = {x: x = 3}’ = {x: x ∈ N and x ≠ 3} Ex1.5,3 Taking the set of natural numbers as the universal set, write down the complements of the following sets: (ix) {x: 2x + 5 = 9} U = {1,2,3,4,5…..} 2x + 5 = 9 2x = 9 – 5 2x = 4 x = ﷐4﷮2﷯ x = 2 {x: 2x + 5 = 9}´ = {x : x = 2}’ = {x: x ∈ N and x ≠ 2} Ex1.5,3(Method 1) Taking the set of natural numbers as the universal set, write down the complements of the following sets: (x) {x: x ≥ 7} U = {1,2,3,4,5…..} {x: x ≥ 7}´ = {x: x ∈ N and x < 7} Ex1.5,3(Method 2) Taking the set of natural numbers as the universal set, write down the complements of the following sets: (x) {x: x ≥ 7} U = {1,2,3,4,5…..} {x: x ≥ 7} = {7, 8, 9, 10, 11, …..} {x: x ≥ 7}´ = {7, 8, 9, 10, 11, …..}’ = U – {7, 8, 9, 10, 11, …..} = {1,2,3,4,5,6,7,8…..} – {7, 8, 9, 10, 11, …..} = {1, 2, 3, 4, 5, 6} = {x: x ∈ N and x < 7} Ex1.5,3(Method 1) Taking the set of natural numbers as the universal set, write down the complements of the following sets: (xi) {x: x ∈ N and 2x + 1 > 10} U = {1,2,3,4,5…..} 2x + 1 > 10 2x > 10 – 1 2x > 9 x > ﷐9﷮2﷯ x > 4.5 {x: x ∈ N and 2x + 1 > 10} = {x: x ∈ N and x > 4.5} {x: x ∈ N and 2x + 1 > 10}’ = {x: x ∈ N and x > 4.5}’ = {x: x ∈ N and x ≤ 4.5} Ex1.5,3(Method 2) Taking the set of natural numbers as the universal set, write down the complements of the following sets: (xi) {x: x ∈ N and 2x + 1 > 10} U = {1,2,3,4,5…..} 2x + 1 > 10 2x > 10 – 1 2x > 9 x > ﷐9﷮2﷯ x > 4.5 {x: x ∈ N and 2x + 1 > 10} = {x: x ∈ N and x > 4.5} = { 5, 6, 7, 8, 9, 10,…..} {x: x ∈ N and 2x + 1 > 10}’ = { 5, 6, 7, 8, 9, 10,…..}’ = U – { 5, 6, 7, 8, 9, 10,…..} = {1,2,3,4,5,6,7….} – { 5, 6, 7, 8, 9, 10,…..} = {1, 2, 3, 4} = {x: x ∈ N and x ≤ 4.5} Thus, {x: x ∈ N and 2x + 1 > 10}’ = {x: x ∈ N and x ≤ 4.5}

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