Chapter 1: Sets Exercise 1.5

Ans: Given that,

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

To find,

$A'$

The complement of set A is the set of all elements of U which are not the elements of A.

$A'=U-A$

$=\left\{ 1,2,3,4,5,6,7,8,9 \right\}-\left\{ 1,2,3,4 \right\}$

$=\left\{ 5,6,7,8,9 \right\}$

$\therefore A'=\left\{ 5,6,7,8,9 \right\}$

Ans: Given that,

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

To find,

$B'$

The complement of set A is the set of all elements of U which are not the elements of A.

$B'=U-B$

$=\left\{ 1,2,3,4,5,6,7,8,9 \right\}-\left\{ 2,4,6,8 \right\}$

$=\left\{ 1,3,5,7,9 \right\}$

$\therefore B'=\left\{ 1,3,5,7,9 \right\}$

Ans: Given that,

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

To find,

$\left( A\cup C \right)'$

The complement of set A is the set of all elements of U which are not the elements of A.

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

$\left( A\cup C \right)'=U-\left( A\cup C \right)$

$=\left\{ 1,2,3,4,5,6,7,8,9 \right\}-\left\{ 1,2,3,4,5,6 \right\}$

$=\left\{ 7,8,9 \right\}$

$\therefore \left( A\cup C \right)'=\left\{ 7,8,9 \right\}$

Ans: Given that,

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

To find,

$\left( A\cup B \right)'$

The complement of set A is the set of all elements of U which are not the elements of A.

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

$\left( A\cup B \right)'=U-A\cup B$

$=\left\{ 1,2,3,4,5,6,7,8,9 \right\}-\left\{ 1,2,3,4

,6,8 \right\}$

$=\left\{ 5,7,9 \right\}$

$\therefore \left( A\cup B \right)'=\left\{ 5,7,9 \right\}$

Ans: Given that,

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

To find,

$\left( A' \right)'$

The complement of set A is the set of all elements of U which are not the elements of A.

$\left( A' \right)'=A$

$=\left\{ 1,2,3,4 \right\}$

$\therefore \left( A' \right)'=\left\{ 1,2,3,4 \right\}$

Ans: Given that,

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

To find,

$\left( B-C \right)'$

The complement of set A is the set of all elements of U which are not the elements of A.

$B-C=\left\{ 2,8 \right\}$

$\left( B-C \right)'=U-\left( B-C \right)$

$=\left\{ 1,2,3,4,5,6,7,8,9 \right\}-\left\{ 2,8 \right\}$

$=\left\{ 1,3,4,5,6,7,9 \right\}$

$\therefore \left( B-C \right)'=\left\{ 1,3,4,5,6,7,9 \right\}$

Ans: Given that,

$U=\left\{ a,b,c,d,e,f,g,h \right\}$

$A=\left\{ a,b,c \right\}$

To find the complement of A

The complement of set A is the set of all elements of U which are not the elements of A.

$A'=U-A$

$=\left\{ a,b,c,d,e,f,g,h \right\}-\left\{ a,b,c \right\}$

$=\left\{ d,e,f,g,h \right\}$

$\therefore $The complement of $A$ is $A'=\left\{ d,e,f,g,h \right\}$

Ans: Given that,

$U=\left\{ a,b,c,d,e,f,g,h \right\}$

$b=\left\{ d,e,f,g \right\}$

To find the complement of B

The complement of set A is the set of all elements of U which are not the elements of A.

$B'=U-B$

$=\left\{ a,b,c,d,e,f,g,h \right\}-\left\{ d,e,f,g \right\}$

$=\left\{ a,b,c,h \right\}$

$\therefore $The complement of $B$ is $B'=\left\{ b,e,c,h \right\}$

Ans: Given that,

$U=\left\{ a,b,c,d,e,f,g,h \right\}$

$C=\left\{ a,c,e,g \right\}$

To find the complement of A

The complement of set A is the set of all elements of U which are not the elements of A.

$C'=U-C$

$=\left\{ a,b,c,d,e,f,g,h \right\}-\left\{ a,c,e,g \right\}$

$=\left\{ b,d,f,h \right\}$

$\therefore $The complement of $C$ is $C'=\left\{ b,d,f,h \right\}$

Ans: Given that,

$U=\left\{ a,b,c,d,e,f,g,h \right\}$

$A=\left\{ f,g,h,a \right\}$

To find the complement of A

The complement of set A is the set of all elements of U which are not the elements of A.

$D'=U-D$

$=\left\{ a,b,c,d,e,f,g,h \right\}-\left\{ f,g,h,a \right\}$

$=\left\{ b,c,d,e \right\}$

$\therefore $The complement of $D$ is $D'=\left\{ b,c,d,e \right\}$

Ans: Given that,

The set of natural number is the universal set

To find the complement of the set of even natural number

The complement of set A is the set of all elements of U which are not the elements of A.

$\therefore \left\{ x:x\text{ is an even natural number} \right\}'=\left\{ x:x\text{ is an odd natural number} \right\}$

Ans: Given that,

The set of natural number is the universal set

To find the complement of the set of odd natural number

The complement of set A is the set of all elements of U which are not the elements of A.

$\therefore \left\{ x:x\text{ is an odd natural number} \right\}'=\left\{ x:x\text{ is an even natural number} \right\}$

Ans: Given that,

The set of natural number is the universal set

To find the complement of the set of positive multiples of $3$

The complement of set A is the set of all elements of U which are not the elements of A.

$\therefore \left\{ x:x\text{ is a positive multiple of 3} \right\}'=\left\{ x:x\in N\text{ and x is not a positive multiple of 3} \right\}$

Ans: Given that,

The set of natural number is the universal set

To find the complement of the set of prime number

The complement of set A is the set of all elements of U which are not the elements of A.

$\therefore \left\{ x:x\text{ is a prime number} \right\}'=\left\{ x:x\text{ is a positive composite number and x=1} \right\}$

Ans: Given that,

The set of natural number is the universal set

To find the complement of the set of natural number divisible by $3$ and $5$

The complement of set A is the set of all elements of U which are not the elements of A.

$\therefore \left\{ x:x\text{ is a number divisible by 3 and 5} \right\}'=$

$\left\{ x:x\text{ is a natural number that is not divisible by 3 or 5} \right\}$

Ans: Given that,

The set of natural number is the universal set

To find the complement of the set of perfect squares.

The complement of set A is the set of all elements of U which are not the elements of A.

$\therefore \left\{ x:x\text{ is a perfect squares} \right\}'=\left\{ x:x\in N\text{ and x is not a perfect square} \right\}$

Ans: Given that,

The set of natural number is the universal set

To find the complement of the set of perfect cube

The complement of set A is the set of all elements of U which are not the elements of A.

$\therefore \left\{ x:x\text{ is a perfect cube} \right\}'=\left\{ x:x\in N\text{ and x is not a perfect cube} \right\}$

Ans: Given that,

The set of natural number is the universal set

To find the complement of $\left\{ x:x+5=8 \right\}$

$x+5=8$

$x=3$

The complement of set A is the set of all elements of U which are not the elements of A.

$\therefore \left\{ x:x+5=8 \right\}'=\left\{ x:x\in N\text{ and x}\ne \text{3} \right\}$

Ans: Given that,

The set of natural number is the universal set

To find the complement of the 

$\left\{ x:2x+5=9 \right\}$

The complement of set A is the set of all elements of U which are not the elements of A.

$2x+5=9$

$2x=4$

$x=2$

$\therefore \left\{ x:2x+5=9 \right\}'=\left\{ x:x\in N\text{ and x}\ne \text{2 } \right\}$

Ans: Given that,

The set of natural number is the universal set

To find the complement of 

$\left\{ x:x\ge 7 \right\}$

The complement of set A is the set of all elements of U which are not the elements of A.

$\therefore {{\left\{ x:x\ge 7 \right\}}^{\prime }}=\left\{ x:x\in N\text{ and x}<\text{7} \right\}$

Ans: Given that,

The set of natural number is the universal set

To find the complement of the 

$\left\{ x:x\in N\text{ and 2x+1 > 10} \right\}$

The complement of set A is the set of all elements of U which are not the elements of A.

$2x+1>10$

$2x>9$

$x>\frac{9}{2}$

$\therefore \left\{ x:x\in N\text{ and 2x+1 > 10} \right\}'$$=\left\{ x:x\in N\text{ and x}\le \frac{9}{2} \right\}$

Ans: Given that.

$U=\left\{ 1,2,3,4,5,6,7,8,9 \right\}$

$A=\left\{ 2,4,6,8 \right\}$

$B=\left\{ 2,3,5,7 \right\}$

To prove that $\left( A\cup B \right)'=A'\cap B'$

$A\cup B=\left\{ 2,4,6,8 \right\}\cup \left\{ 2,3,5,7 \right\}$

$=\left\{ 2,3,4,5,6,7,8 \right\}$

$\left( A\cup B \right)'=U=A\cup B$

$=\left\{ 1,9 \right\}$

$A'=U-A$

$=\left\{ 1,3,5,7,9 \right\}$

$B'=U-B$

$=\left\{ 1,4,6,8,9 \right\}$

$A'\cap B'=\left\{ 1,3,5,7,9 \right\}\cap \left\{ 1,4,6,8,9 \right\}$

$=\left\{ 1,9 \right\}$

Hence it has been proved that $\left( A\cup B \right)'=A'\cap B'$

Ans: Given that.

$U=\left\{ 1,2,3,4,5,6,7,8,9 \right\}$

$A=\left\{ 2,4,6,8 \right\}$

$B=\left\{ 2,3,5,7 \right\}$

To prove that $\left( A\cap B \right)'=A'\cup B'$

$A\cap B=\left\{ 2,4,6,8 \right\}\cap \left\{ 2,3,5,7 \right\}$

$=\left\{ 2 \right\}$

$\left( A\cap B \right)'=U-A\cap B$

$=\left\{ 1,3,4,5,6,7,8,9 \right\}$

$A'=U-A$

$=\left\{ 1,3,5,7,9 \right\}$

$B'=U-B$

$=\left\{ 1,4,6,8,9 \right\}$

$A'\cup B'=\left\{ 1,3,5,7,9 \right\}\cup \left\{ 1,4,6,8,9 \right\}$

$=\left\{ 1,3,4,5,6,7,8,9 \right\}$

Hence it has been proved that $\left( A\cap B \right)'=A'\cup B'$

Ans: To draw the Venn diagram for $\left( A\cup B \right)'$

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Ans: To draw the Venn diagram for $A'\cap B'$

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Ans: To draw the Venn diagram for $\left( A\cap B \right)'$

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Ans: To draw the Venn diagram for $A'\cup B'$

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Ans: Given that,

$U$ is the set of all triangles in the plane

$A=$Set of triangles different form $60{}^\circ $

To find $A'$

The complement of set A is the set of all elements of U which are not the elements of A.

$A'=U-A$

$=$Set of all equilateral triangles

$\therefore A'$ is the set of all equilateral triangles

Ans: To fill the blanks given in the statement

The union of the set and its complement is the universal set

$\therefore A\cup A'=U$

Ans: To fill the blanks given in the statement

We know that,

$\varnothing '\cap A=U\cap A=A$

$\therefore \varnothing '\cap A=A$

Ans: To fill the blanks given in the statement

The intersection of the set and its complement is an empty set.

$\therefore A\cap A'=\varnothing $

Ans: To fill the blanks given in the statement

We know that,

$\varnothing \cap A=U'\cap A=\varnothing $

$\therefore U'\cap A=\varnothing $