NCERT Solutions For Class 11 Maths Chapter 1 Exercise 1.4

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Chapter 1 Ex.1.4 Question 1

Find the union of each of the following pairs of sets:

(i) \(X = \left\{ {1,3,5} \right\}\;\;\;\;\;\;\;\;\;\;\;Y = \left\{ {1,2,3} \right\}\)

(ii) \(A=\left\{ a,~e,~i,~o,~u \right\}\ \ \ \ \ B=\left\{ a,~b,~c \right\}\)

(iii) \(A =\) {\(x:x\) is a natural number and multiple of \(3\)}

     \(B =\) {\(x:x\) is a natural number less than \(6\)}

(iv) \(A=\) {\(x:x\) is a natural number and \(1 \lt x \le 6\) }

     \(B =\) {\(x:x\) is a natural number and \(6 \lt x \lt 10\) }

(v) \(A = \left\{ {1,2,3} \right\}\;\;\;\;\;\;B = \phi \)

Solution

(i) \(X = \left\{ {1,3,5} \right\}\;\;\;\;\;\;\;\;\;\;\;Y = \left\{ {1,2,3} \right\}\)

 \(X \cup Y = \left\{ {1,2,3,5} \right\}\)

(ii) \(A=\left\{ a,~e,~i,~o,~u \right\}\ \ \ \ \ B=\left\{ a,~b,~c \right\}\)

\(A\cup B=\left\{ a,~b,~c,~e,~i,~o,~u \right\}\)

(iii) \(A =\) {\(x:x\) is a natural number and multiple of \(3\)}

\({\rm{A = }}\left\{ {3,6,9 \ldots } \right\}\)

     \(B =\) {\(x:x\) is a natural number less than \(6\)}

\(B = \left\{ {1,2,3,4,5,6} \right\}\)

\(A \cup B = \left\{ {1,2,4,5,3,6,9,12 \ldots } \right\}\)

Therefore, \(A \cup B =\) { \( x:x = 1,2,4,5 \) or a multiple of 3}

(iv) \(A =\) {\(x:x\) is a natural number and \(1 \lt x \le 6\) }

\(A = \left\{ {2,3,4,5,6} \right\}\)

      \(B =\) {\(x:x\) is a natural number and \(6 \lt x \lt 10\) }

\(B = \left\{ {7,8,9} \right\}\)

\(A \cup B = \left\{ {2,3,4,5,6,7,8,9} \right\}\)

Therefore, \(A \cup B = \left\{ {x:x \in N\;\;{\rm{and}}\;\;1 \lt x \lt 10} \right\}\)

(v) \(A = \left\{ {1,2,3} \right\}\;\;\;\;\;\;B = \phi \)

\(A \cup B = \left\{ {1,2,3} \right\}\)

Chapter 1 Ex.1.4 Question 2

Let \(A=\left\{ a,~b \right\},\ B=\left\{ a,~b,~c \right\}\). Is \(A \subset B\)? What is \(A \cup B\)?

Solution

Here, \(A=\left\{ a,~b \right\}\) and \(B=\left\{ a,~b,~c \right\}\)

We see, \(B\) consists all the elements of \(A\)

Hence, Yes \(A \subset B\).

\[A\cup B=\left\{ a,~b,~c \right\}\]

Chapter 1 Ex.1.4 Question 3

If \(A\) and \(B\) are two sets such that \(A \subset B\), then what is \(A \cup B\)?

Solution

If \(A\) and \(B\) are two sets such that \(A \subset B\), that means B consists all the elements of \(A\)

Then, \(A \cup B = B\).

Chapter 1 Ex.1.4 Question 4

If \(A = \left\{ {1,2,3,4} \right\},\;\;B = \left\{ {3,4,5,6} \right\},\;\;C = \left\{ {5,6,7,8} \right\}\) and \(D = \left\{ {7,8,9,10} \right\}\); find

(i) \(A \cup B\)

(ii) \(A \cup C\)

(iii) \(B \cup C\)

(iv) \(B \cup D\)

(v) \(A \cup B \cup C\)

(vi) \(A \cup B \cup D\)

(vii) \(B \cup C \cup D\)

Solution

\(A = \left\{ {1,2,3,4} \right\},\;\;B = \left\{ {3,4,5,6} \right\},\;\;C = \left\{ {5,6,7,8} \right\}\) and \(D = \left\{ {7,8,9,10} \right\}\)

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

(ii) \(A \cup C = \left\{ {1,2,3,4,5,6,7,8} \right\}\)

(iii) \(B \cup C = \left\{ {3,4,5,6,7,8} \right\}\)

(iv) \(B \cup D = \left\{ {3,4,5,6,7,8,9,10} \right\}\)

(v) \(A \cup B \cup C = \left\{ {1,2,3,4,5,6,7,8} \right\}\)

(vi) \(A \cup B \cup D = \left\{ {1,2,3,4,5,6,7,8,9,10} \right\}\)

(vii) \(B \cup C \cup D = \left\{ {3,4,5,6,7,8,9,10} \right\}\)

Chapter 1 Ex.1.4 Question 5

Find the intersection of each pair of sets for the following.

(i) \(X = \left\{ {1,3,5} \right\}\;\;\;\;\;\;\;\;\;\;\;Y = \left\{ {1,2,3} \right\}\)

(ii) \(A=\left\{ a,~e,~i,~o,~u \right\}\ \ \ \ \ B=\left\{ a,~b,~c \right\}\)

(iii) \(A =\) {\(x:x\) is a natural number and multiple of \(3\)}

\(B =\) {\(x:x\) is a natural number less than\( 6\)}

(iv)\(A =\) {\(x:x\) is a natural number and \(1 \lt x \le 6\)}

\(B =\){\(x:x\) is a natural number and \(6 \lt x \lt 10\)}

(v) \(A = \left\{ {1,2,3} \right\}\;\;\;\;\;\;B = \phi \)

Solution

(i) \(X = \left\{ {1,3,5} \right\}\;\;\;\;\;\;\;\;\;\;\;Y = \left\{ {1,2,3} \right\}\)

\[X \cap Y = \left\{ {1,3} \right\}\]

(ii) \(A=\left\{ a,~e,~i,~o,~u \right\}\ \ \ \ \ B=\left\{ a,~b,~c \right\}\)

\[A \cap B = \left\{ a \right\}\]

(iii) \(A =\) {\(x:x\) is a natural number and multiple of \(3\)}

     \(B =\) {\(x:x\) is a natural number less than \(6\)}

\[A \cap B = \left\{ 3 \right\}\]

(iv) \(A =\) {\(x:x\) is a natural number and \(1 \lt x \le 6\)}

      \(B =\) {\(x:x\) is a natural number and \(6 \lt x \lt 10\)}

\[A \cap B = \phi \]

(v) \(A = \left\{ {1,2,3} \right\}\;\;\;\;\;\;B = \phi \)

\[A \cap B = \phi \]

Chapter 1 Ex.1.4 Question 6

If \(A = \left\{ {3,5,7,9,11} \right\},\;B = \left\{ {7,9,11,13} \right\},\;C = \left\{ {11,13,15} \right\}\) and \(D = \left\{ {15,17} \right\}\); find

(i) \(A \cap B\)

(ii) \(B \cap C\)

(iii) \(A \cap C \cap D\)

(iv) \(A \cap C\)

(v) \(B \cap D\)

(vi) \(A \cap \left( {B \cup C} \right)\)

(vii) \(A \cap D\)

(viii) \(A \cap \left( {B \cup D} \right)\)

(ix) \(\left( {A \cap B} \right) \cap \left( {B \cup C} \right)\)

(x) \(\left( {A \cup D} \right) \cap \left( {B \cup C} \right)\)

Solution

(i) \(A \cap B = \left\{ {7,9,11} \right\}\)

(ii) \(B \cap C = \left\{ {11,13} \right\}\)

(iii) \(A \cap C \cap D = \left\{ {A \cap C} \right\} \cap D = \left\{ {11} \right\} \cap \left\{ {15,17} \right\} = \phi \)

(iv) \(A \cap C = \left\{ {11} \right\}\)

(v) \(B \cap D = \phi \)

(vi) \(A \cap \left( {B \cup C} \right) = \left( {A \cap B} \right) \cup \left( {A \cap C} \right) = \left\{ {7,9,11} \right\} \cup \left\{ {11} \right\} = \left\{ {7,9,11} \right\}\)

(vii) \(A \cap D = \phi \)

(viii) \(A\cap \left( B\cup D \right)=\left( A\cap B \right)\cup \left( A\cap D \right)=\left\{ 7,9,11 \right\}~\cup \phi =\left\{ 7,9,11 \right\}\)

(ix) \(\left( {A \cap B} \right) \cap \left( {B \cup C} \right) = \left\{ {7,9,11} \right\} \cap \left\{ {7,9,11,13,15} \right\} = \left\{ {7,9,11} \right\}\)

(x) \(\left( {A \cup D} \right) \cap \left( {B \cup C} \right) = \left\{ {3,5,7,9,11,15,17} \right\} \cap \left\{ {7,9,11,13,15} \right\} = \left\{ {7,9,11,15} \right\}\)

Chapter 1 Ex.1.4 Question 7

If \(A =\) {\(x:x\) is a natural number}, \(B =\) {\(x:x\) is an even natural number}

 \(C =\) {\(x:x\) is an odd natural number} and \(D =\) {\(x:x\) is a prime number}, find

(i) \(A \cap B\)

(ii) \(A \cap C\)

(iii) \(A \cap D\)

(iv) \(B \cap C\)

(v) \(B \cap D\)

(vi) \(C \cap D\)

Solution

\(A =\) {\(x:x\) is a natural number} \( = \left\{ {1,2,3,4,5 \ldots } \right\}\)

\(B =\) {\(x:x\) is an even natural number} \( = \left\{ {2,4,6,8 \ldots } \right\}\)

\(C =\) {\(x:x\) is an odd natural number} \( = \left\{ {1,3,5,7,9 \ldots } \right\}\)

\(D =\) {\(x:x\) is a prime number} \( = \left\{ {2,3,5,7 \ldots } \right\}\)

(i) \(A \cap B = \) {\(x:x\) is a even natural number} \(= B\)

(ii) \(A \cap C = \) {\(x:x\) is an odd natural number} \(= C\)

(iii) \(A \cap D = \) {\(x:x\) is a prime number} \(= D\)

(iv) \(B \cap C = \phi \)

(v) \(B \cap D = \left\{ 2 \right\}\)

(vi) \(C \cap D = \){\(x:x\) is an odd prime number}

Chapter 1 Ex.1.4 Question 8

Which of the following pairs of sets are disjoint

(i) \(\left\{ {1,2,3,4} \right\}\) and {\(x:x\) is a natural number and \(4 \le x \le 6\)}

(ii) \(\left\{ a,~e,~i,~o,~u \right\}\) and \(\left\{ c,~d,~e,~f \right\}\)

(iii) {\(x:x\) is an even integer} and {\(x:x\) is an odd integer}

Solution

(i) \(\left\{ {1,2,3,4} \right\}\)

{\(x:x\) is a natural number and \(4 \le x \le 6\)}\( = \left\{ {4,5,6} \right\}\)

Now, \(\left\{ {1,2,3,4} \right\} \cap \left\{ {4,5,6} \right\} = \left\{ 4 \right\}\)

Therefore, this pair of sets is not disjoint.

(ii) \(\left\{ a,~e,~i,~o,~u \right\}\cap \left\{ c,~d,~e,~f \right\}=\left\{ e \right\}\)

Therefore, this pair of sets are not disjoint.

(iii) {\(x:x\) is an even integer} \( \cap \) {\(x:x\) is an odd integer} \( = \phi \)

Therefore, this pair of sets is disjoint.

Chapter 1 Ex.1.4 Question 9

If \(A = \left\{ {3,6,9,12,15,18,21} \right\},\) \(B = \left\{ {4,8,12,16,20} \right\}\), \(C = \left\{ {2,4,6,8,10,12,14,16} \right\},\) \(D = \left\{ {5,10,15,20} \right\};\) find

(i) \(A - B\)

(ii) \(A - C\)

(iii) \(A - D\)

(iv) \(B - A\)

(v) \(C - A\)

(vi) \(D - A\)

(vii) \(B - C\)

(viii) \(B - D\)

(ix) \(C - B\)

(x) \(D - B\)

(xi) \(C-D\)

(xii) \(D - C\)

Solution

(i) \(A - B = \left\{ {3,6,9,15,18,21} \right\}\)

(ii) \(A-C = \left\{ {3,9,15,18,21} \right\}\)

(iii) \(A-D = \left\{ {3,6,9,12,18,21} \right\}\)

(iv) \(B - A = \left\{ {4,8,16,20} \right\}\)

(v) \(C-A = \left\{ {2,4,8,10,14,16} \right\}\)

(vi) \(D - A = \left\{ {5,10,20} \right\}\)

(vii) \(B-C = \left\{ {20} \right\}\)

(viii) \(B - D = \left\{ {4,8,12,16} \right\}\)

(ix) \(C - B = \left\{ {2,6,10,14} \right\}\)

(x) \(D-B = \left\{ {5,10,15} \right\}\)

(xi) \(C - D = \left\{ {2,4,6,8,12,14,16} \right\}\)

(xii) \(D - C = \left\{ {5,15,20} \right\}\)

Chapter 1 Ex.1.4 Question 10

If \(X=\left\{ a,~b,~c,~d \right\}\) and \(Y=\left\{ f,~b,~d,g \right\}\) , find

(i) \(X - Y\)

(ii) \(Y-X\)

(iii) \(X \cap Y\)

Solution

(i) \(X-Y=\left\{ a,~c \right\}\)

(ii) \(Y-X = \left\{ {f,g} \right\}\)

(iii) \(X\cap Y=\left\{ b,~d \right\}\)

Chapter 1 Ex.1.4 Question 11

If \(R\) is the set of real numbers and \(Q\) is the set of rational numbers, then what is \(R – Q\)?

Solution

\(R:\) set of real numbers

\(Q:\) set of rational numbers

Therefore, \(R – Q\) is a set of irrational numbers.

Chapter 1 Ex.1.4 Question 12

State whether each of the following statement is true or false. Justify your answer.

(i) \(\left\{ {2,3,4,5} \right\}\) and \(\left\{ {3,6} \right\}\) are disjoint sets.

(ii) \(\left\{ a,e,i,o,u \right\}\) and \(\left\{ a,b,c,d \right\}\) are disjoint sets.

(iii) \(\left\{ 2,6,10,14 \right\}\) and \(\left\{ 3,7,11,15 \right\}\) are disjoint sets.

(iv) \(\left\{ 2,6,10 \right\}\) and \(\left\{ {3,7,11} \right\}\) are disjoint sets.

Solution

(i) False

Since, \(3 \in \left\{ {2,3,4,5} \right\}\) and \(3 \in \left\{ {3,6} \right\}\)

Therefore, \(\left\{ {2,3,4,5} \right\} \cap \left\{ {3,6} \right\} = \left\{ 3 \right\}\)

(ii) False

Since, \(a \in \left\{ {a,e,i,o,u} \right\}\)and \(a \in \left\{ {a,b,c,d} \right\}\)

Therefore, \(\left\{ {a,e,i,o,u} \right\} \cap \left\{ {a,b,c,d} \right\} = \left\{ a \right\}\)

(iii) True

Since, \(\left\{ {2,6,10,14} \right\} \cap \left\{ {3,7,11,15} \right\} = \phi \)

(iv) True

Since, \(\left\{ {2,6,10} \right\} \cap \left\{ {3,7,11} \right\} = \phi \)

  
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