NCERT Solutions For Class 11 Maths Chapter 1 Exercise 1.5

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Chapter 1 Ex.1.5 Question 1

Let \(U = \left\{ {1,2,3,4,5,6,7,8,9} \right\},\;A = \left\{ {1,2,3,4} \right\},\;B = \left\{ {2,4,6,8} \right\}\) and \(C = \left\{ {3,4,5,6} \right\}\). Find

(i) \(A^\prime \)

(ii) \({B}^\prime \)

(iii) \(\left( {A \cup C} \right)^\prime \)

(iv)\(\left( {A \cup B} \right)^\prime \)

(v) \(\left( {A\prime } \right)\prime \)

(vi) \(\left( {B-C} \right)^\prime \)

Solution

\(U = \left\{1,2,3,4,5,6,7,8,9\right\},\;A = \left\{ 1,2,3,4 \right\},\;B = \left\{ {2,4,6,8} \right\}\) and \(C = \left\{ 3,4,5,6 \right\}\)

(i) \(A\prime = \left\{ {5,6,7,8,9} \right\}\)

(ii) \(B\prime = \left\{ {1,3,5,7,9} \right\}\)

(iii) \(A \cup C = \left\{ {1,2,3,4,5,6} \right\}\)

Therefore, \(\left( {A \cup C} \right)\prime = \left\{ {7,8,9} \right\}\)

(iv) \(A \cup B = \left\{ {1,2,3,4,6,8} \right\}\)

Therefore, \(\left( {A \cup C} \right)\prime = \left\{ {7,8,9} \right\}\)

(v)\((A^\prime)^\prime=A=\left\{ {1,2,3,4} \right\} \)

(vi) \(B-C = \left\{ {2,8} \right\}\)

Therefore, \(\left( {B-C} \right)\prime = \left\{ {1,3,4,5,6,7,9} \right\}\)

Chapter 1 Ex.1.5 Question 2

If \(U = \left\{ {a,b,c,d,e,f,g,h} \right\}\) , find the complements of the following sets:

(i) \(A = \left\{ a,b,c \right\}\)

(ii) \(B = \left\{ d,e,f,g \right\}\)

(iii) \(C = \left\{a,c,e,g \right\}\)

(iv) \(D = \left\{f,g,h,a\right\}\)

Solution

(i) \({A}^\prime = \left\{ d,e,f,g,h \right\}\)

(ii) \({B}^\prime = \left\{ a,b,c,h \right\}\)

(iii) \({C}^\prime = \left\{ b,d,f,h \right\}\)

(iv) \({D}^\prime = \left\{ b,c,d,e \right\}\)

Chapter 1 Ex.1.5 Question 3

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}

(ii) {\(x:x\) is an odd natural number}

(iii) {\(x:x\) is a positive multiple of \(3\)}

(iv) {\(x:x\) is a prime number}

(v) {\(x:x\) is a natural number divisible by \(3\) and \(5\)}

(vi) {\(x:x\) is a perfect square}

(vii) {\(x:x\) is perfect cube}

(viii) \(\left\{x:x + 5 = 8\right\}\)

(ix) \(\left\{x:2x + 5 = 9 \right\}\)

(x) \(\left\{ x:x \ge 7 \right\}\)

(xi) \(\left\{ {x:x \in N\;{\rm{and}}\;2x + 1 > 10} \right\}\)

Solution

\(U = N\): Set of natural numbers

(i) {\(x:x\) is an even natural number} ´ \(=\) {\(x:x\) is an odd natural number}

(ii) {\(x:x\) is an odd natural number} ´ \(=\) {\(x:x\) is an even natural number}

(iii) {\(x:x\) is a positive multiple of \(3\)} ´ \(=\) {\(x:x\) \(∈ N\) and \(x\) is not a multiple of \(3\)}

(iv) {\(x:x\) is a prime number} ´ \(=\){ \(x:x\) is a positive composite number and \(x~=1\)}

(v) {\(x:x\) is a natural number divisible by \(3\) and \(5\)} ´ \(=\) {\(x:x\) is a natural number that is not divisible by \(3\) or \(5\)}

(vi) {\(x:x\) is a perfect square}´ \(=\) { \({x:x} \in N\) and \(x\) is not a perfect square}

(vii) {\(x:x\) is a perfect cube}´ \(=\) {\(x:x \in N\) and \(x\) is not a perfect cube}

(viii) \({\left\{ {x:x + 5 = 8} \right\}^\prime } = \left\{ {x:x \in N\;{\rm{and}}\;x \ne 3} \right\}\)

(ix) \({\left\{ {x:2x + 5 = 9} \right\}^\prime } = \left\{ {x:x \in N\;{\rm{and}}\;x \ne 2} \right\}\)

(x) \({\left\{ {x:x \ge 7} \right\}^\prime } = \left\{ {x:x \in N\;{\rm{and}}\;x < 7} \right\}\)

(xi) {\(x:x ∈ N\) and \(2x+1>10\) }\(^\prime=\) \(\left\{ {x:x ∈ N \text{and} \,x<\frac{9}{2} } \right\}\)

Chapter 1 Ex.1.5 Question 4

If \(U = \left\{ {1,2,3,4,5,6,7,8,9} \right\},\;A = \left\{ {2,4,6,8} \right\}\) and \(B=\left\{ 2,3,5,7 \right\}\). Verify that

(i)\({\left( {A \cup B} \right)^\prime } = A' \cap B'\)

(ii) \({{\left( A\cap B \right)}^{\prime }}={A}'\cup {B}'\)

Solution

(i) \(U = \left\{ {1,2,3,4,5,6,7,8,9} \right\},\;A = \left\{ {2,4,6,8} \right\}\) and \(B = \left\{ {2,3,5,7} \right\}\)

\[\begin{align}A \cup B&= \left\{ {2,3,4,5,6,7,8} \right\}\\{\left( {A \cup B} \right)^\prime }&= \left\{ {1,9} \right\}\\A'&= \left\{ {1,3,5,7,9} \right\}\\B'&= \left\{ {1,4,6,8,9} \right\}\\A' \cap B'&= \left\{ {1,9} \right\}\\{\left( {A \cup B} \right)^\prime } &= A' \cap B'\end{align}\]

(ii) \(U = \left\{ {1,2,3,4,5,6,7,8,9} \right\},\;A = \left\{ {2,4,6,8} \right\}\) and \(B = \left\{ {2,3,5,7} \right\}\)

\[\begin{align}A \cap B&= \left\{ 2 \right\}\\{\left( {A \cap B} \right)^\prime } &= \left\{ {1,3,4,5,6,7,8,9} \right\}\\A' \cup B' &= \left\{ {1,3,4,5,6,7,8,9} \right\}\\{\left( {A \cap B} \right)^\prime } &= A' \cup B'\end{align}\]

Chapter 1 Ex.1.5 Question 5

Draw appropriate Venn diagram for each of the following:

(i) \({\left( {A \cup B} \right)^\prime }\)

(ii) \(A' \cap B'\)

(iii) \({\left( {A \cap B} \right)^\prime }\)

(iv) \(A' \cup B'\)

Solution

(i) \({\left( {A \cup B} \right)^\prime }\)

(ii) \(A' \cap B'\)

(iii) \({\left( {A \cap B} \right)^\prime }\)

(iv) \(A' \cup B'\)

Chapter 1 Ex.1.5 Question 6

Let \(U\) be the set of all triangles in a plane. If \(A\) is the set of all triangles with at least one angle different from \(60^\circ\), what is \(A\)′?

Solution

\(A′\) is the set of all equilateral triangles.

Chapter 1 Ex.1.5 Question 7

Fill in the blanks to make each of the following a true statement:

(i) \(A \cup A' =\underline{\;\;\;\;}\)

(ii) \(\phi \prime \cap A =\underline{\;\;\;\;}\)

(iii) \(A \cap A' =\underline{\;\;\;\;}\)

(iv) \(U' \cap A =\underline{\;\;\;\;} \)

Solution

(i) \(A \cup A' = \underline{\;U\;} \)

(ii) \(\phi \prime \cap A = \underline{\;U \cap A = A\; }\)

(iii) \(A \cap A' = \underline{\;\phi\;} \)

(iv) \(U' \cap A = \underline{\;\phi \cap A = \;\phi \;}\)

  
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