Q1. Find the union of each of the following pairs of sets :
(i) X = {1, 3, 5} Y = {1, 2, 3}
(ii) A = {a, e, i, o, u} B = {a, b, c}
(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 < x ≤6 } B={x : x is a natural number and 6 < x < 10 }
(v) A={1, 2, 3}, B=φ

Solution

Solution: The union of two sets is the set containing all elements which belong to either of the two sets or to both of them, without repetition.

\( \begin{aligned} \text{(i)}\;& X=\{1,3,5\} \\ & Y=\{1,2,3\} \\ & X\cup Y=\{1,2,3,5\}. \end{aligned} \)

\( \begin{aligned} \text{(ii)}\;& A=\{a,e,i,o,u\} \\ & B=\{a,b,c\} \\ & A\cup B=\{a,b,c,e,i,o,u\}. \end{aligned} \)

\( \begin{aligned} \text{(iii)}\;& A=\{3,6,9,\ldots\} \\ & B=\{1,2,3,4,5\} \\ & A\cup B=\{1,2,3,4,5,6,9,\ldots\}. \end{aligned} \)

\( \begin{aligned} \text{(iv)}\;& A=\{2,3,4,5,6\} \\ & B=\{7,8,9\} \\ & A\cup B=\{2,3,4,5,6,7,8,9\}. \end{aligned} \)

\( \begin{aligned} \text{(v)}\;& A=\{1,2,3\} \\ & B=\varnothing \\ & A\cup B=\{1,2,3\}. \end{aligned} \)

Q2. Let A = { a, b }, B = {a, b, c}. Is A ⊂ B ? What is A ∪ B ?

Solution

Solution: If one set is a subset of another, then every element of the smaller set already belongs to the larger set. Hence, their union will be the larger set itself.

\( \begin{aligned} A &= \{a, b\} \\ B &= \{a, b, c\} \\ A &\subset B \\ \Rightarrow A \cup B &= \{a, b, c\}. \end{aligned} \)

Thus, when \(A \subset B\), the union of \(A\) and \(B\) is equal to \(B\).

Q3. If A and B are two sets such that A ⊂ B, then what is A ∪ B ?

Solution

Solution: If a set \(A\) is a subset of a set \(B\), then every element of \(A\) already belongs to \(B\). Therefore, the union of \(A\) and \(B\) does not add any new elements to \(B\).

\( \begin{aligned} \text{If } A \subset B,\\ A \cup B = B. \end{aligned} \)

Q4. If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8 }and D = { 7, 8, 9, 10 }; find
(i) A ∪ B
(ii) A ∪ C
(iii) B ∪ C
(iv) B ∪ D
(v) A ∪ B ∪ C
(vi) A ∪ B ∪ D
(vi) B ∪ C ∪ D

Solution

Solution: The union of two or more sets is the set containing all distinct elements that belong to at least one of the given sets.

\( \begin{aligned} A &= \{1,2,3,4\} \\ B &= \{3,4,5,6\} \\ C &= \{5,6,7,8\} \\ D &= \{7,8,9,10\} \end{aligned} \)

\( \begin{aligned} \text{(i)}\;& A \cup B = \{1,2,3,4,5,6\} \\ \text{(ii)}\;& A \cup C = \{1,2,3,4,5,6,7,8\} \\ \text{(iii)}\;& B \cup C = \{3,4,5,6,7,8\} \\ \text{(iv)}\;& B \cup D = \{3,4,5,6,7,8,9,10\} \end{aligned} \)

\( \begin{aligned} \text{(v)}\;& A \cup B \cup C = (A \cup B) \cup C \\ &= \{1,2,3,4,5,6\} \cup \{5,6,7,8\} \\ &= \{1,2,3,4,5,6,7,8\} \end{aligned} \)

\( \begin{aligned} \text{(vi)}\;& A \cup B \cup D = (A \cup B) \cup D \\ &= \{1,2,3,4,5,6\} \cup \{7,8,9,10\} \\ &= \{1,2,3,4,5,6,7,8,9,10\} \end{aligned} \)

\( \begin{aligned} \text{(vii)}\;& B \cup C \cup D = (B \cup C) \cup D \\ &= \{3,4,5,6,7,8\} \cup \{7,8,9,10\} \\ &= \{3,4,5,6,7,8,9,10\} \end{aligned} \)

Q5. Find the intersection of each pair of sets of question 1 above.

Solution

Solution: The intersection of two sets consists of all elements that are common to both sets.

\( \begin{aligned} \text{(i)}\;& X=\{1,3,5\} \\ & Y=\{1,2,3\} \\ & X \cap Y=\{1,3\}. \end{aligned} \)

\( \begin{aligned} \text{(ii)}\;& A=\{a,e,i,o,u\} \\ & B=\{a,b,c\} \\ & A \cap B=\{a\}. \end{aligned} \)

\( \begin{aligned} \text{(iii)}\;& A=\{3,6,9,12,\ldots\} \\ & B=\{1,2,3,4,5\} \\ & A \cap B=\{3\}. \end{aligned} \)

\( \begin{aligned} \text{(iv)}\;& A=\{2,3,4,5,6\} \\ & B=\{7,8,9\} \\ & A \cap B=\varnothing. \end{aligned} \)

\( \begin{aligned} \text{(v)}\;& A=\{1,2,3\} \\ & B=\varnothing \\ & A \cap B=\varnothing. \end{aligned} \)

Q6. If A = { 3, 5, 7, 9, 11 }, B = {7, 9, 11, 13}, C = {11, 13, 15}and D = {15, 17}; find
(i) A ∩ B
(ii) B ∩ C
(iii) A ∩ C ∩ D
(iv) A ∩ C
(v) B ∩ D
(vi) A ∩ (B ∪ C)
(vii) A ∩ D
(viii) A ∩ (B ∪ D)
(ix) ( A ∩ B ) ∩ ( B ∪ C )
(x) ( A ∪ D) ∩ ( B ∪ C)

Solution

Solution: The intersection of sets contains only those elements which are common to all the sets involved in the operation.

\( \begin{aligned} A &= \{3,5,7,9,11\} \\ B &= \{7,9,11,13\} \\ C &= \{11,13,15\} \\ D &= \{15,17\} \end{aligned} \)

\( \begin{aligned} \text{(i)}\;& A \cap B = \{7,9,11\} \\ \text{(ii)}\;& B \cap C = \{11,13\} \end{aligned} \)

\( \begin{aligned} \text{(iii)}\;& A \cap C \cap D = (A \cap C) \cap D \\ &= \{11\} \cap \{15,17\} \\ &= \varnothing \end{aligned} \)

\( \begin{aligned} \text{(iv)}\;& A \cap C = \{11\} \\ \text{(v)}\;& B \cap D = \varnothing \end{aligned} \)

\( \begin{aligned} \text{(vi)}\;& A \cap (B \cup C) \\ &= \{3,5,7,9,11\} \cap \{7,9,11,13,15\} \\ &= \{7,9,11\} \end{aligned} \)

\( \begin{aligned} \text{(vii)}\;& A \cap D = \varnothing \end{aligned} \)

\( \begin{aligned} \text{(viii)}\;& A \cap (B \cup D) \\ &= \{3,5,7,9,11\} \cap \{7,9,11,13,15,17\} \\ &= \{7,9,11\} \end{aligned} \)

\( \begin{aligned} \text{(ix)}\;& (A \cap B) \cap (B \cup C) \\ &= \{7,9,11\} \cap \{7,9,11,13,15\} \\ &= \{7,9,11\} \end{aligned} \)

\( \begin{aligned} \text{(x)}\;& (A \cup D) \cap (B \cup C) \\ &= \{3,5,7,9,11,15,17\} \cap \{7,9,11,13,15\} \\ &= \{7,9,11,15\} \end{aligned} \)

Q7. If A = {x : x is a natural number }, B = {x : x is an even natural number} C = {x : x is an odd natural number}andD = {x : x is a prime number }, find
(i) A ∩ B
(ii) A ∩ C
(iii) A ∩ D
(iv) B ∩ C
(v) B ∩ D
(vi) C ∩ D

Solution

Solution: The intersection of two sets consists of all elements which are common to both sets.

\( \begin{aligned} A &= \{1,2,3,4,\ldots\} \\ B &= \{2,4,6,8,\ldots\} \\ C &= \{1,3,5,7,\ldots\} \\ D &= \{2,3,5,7,11,\ldots\} \end{aligned} \)

\( \begin{aligned} \text{(i)}\;& A \cap B = \{2,4,6,\ldots\} = B \end{aligned} \)

\( \begin{aligned} \text{(ii)}\;& A \cap C = \{1,3,5,7,\ldots\} = C \end{aligned} \)

\( \begin{aligned} \text{(iii)}\;& A \cap D = \{2,3,5,7,11,\ldots\} = D \end{aligned} \)

\( \begin{aligned} \text{(iv)}\;& B \cap C = \varnothing \end{aligned} \)

\( \begin{aligned} \text{(v)}\;& B \cap D = \{2\} \end{aligned} \)

\( \begin{aligned} \text{(vi)}\;& C \cap D = \{3,5,7,11,\ldots\} \end{aligned} \)

Q8.Which of the following pairs of sets are disjoint
(i) {1, 2, 3, 4} and {x : x is a natural number and 4 ≤ x ≤ 6 }
(ii) { a, e, i, o, u } and { c, d, e, f }
(iii) {x : x is an even integer } and {x : x is an odd integer}

Solution

Solution: Two sets are said to be disjoint if they have no element in common, that is, if their intersection is the empty set.

\( \begin{aligned} \text{(i)}\;& \{1,2,3,4\} \text{ and } \{x : x \text{ is a natural number and } 4 \le x \le 6\}. \\ &\text{The second set is } \{4,5,6\}. \\ &\text{Since } 4 \text{ is common to both sets,} \\ &\Rightarrow \text{these sets are not disjoint.} \end{aligned} \)

\( \begin{aligned} \text{(ii)}\;& \{a,e,i,o,u\} \text{ and } \{c,d,e,f\}. \\ &\text{The element } e \text{ is common to both sets.} \\ &\Rightarrow \text{these sets are not disjoint.} \end{aligned} \)

\( \begin{aligned} \text{(iii)}\;& \{x : x \text{ is an even integer}\} \text{ and } \{x : x \text{ is an odd integer}\}. \\ &\text{No integer can be both even and odd at the same time.} \\ &\Rightarrow \text{the intersection is } \varnothing, \text{ so these sets are disjoint.} \end{aligned} \)

Hence, only the pair given in (iii) consists of disjoint sets.

Q9. If A = {3, 6, 9, 12, 15, 18, 21},
B = { 4, 8, 12, 16, 20 },
C = { 2, 4, 6, 8, 10, 12, 14, 16 },
D = {5, 10, 15, 20 }; find
(i) A – B
(ii) A – C
(iii) A – D
(iv) B – A
(v) C – A
(vi) D – A
(vii) B –
(viii) B – D
(ix) C – B
(x) D – B
(xi) C – D
(xii) D – C

Solution

Solution: The difference of two sets \(A - B\) consists of all those elements of \(A\) which do not belong to \(B\).

\( \begin{aligned} A &= \{3,6,9,12,15,18,21\} \\ B &= \{4,8,12,16,20\} \\ C &= \{2,4,6,8,10,12,14,16\} \\ D &= \{5,10,15,20\} \end{aligned} \)

\( \begin{aligned} \text{(i)}\;& A - B = \{3,6,9,15,18,21\} \end{aligned} \)

\( \begin{aligned} \text{(ii)}\;& A - C = \{3,9,15,18,21\} \end{aligned} \)

\( \begin{aligned} \text{(iii)}\;& A - D = \{3,6,9,12,18,21\} \end{aligned} \)

\( \begin{aligned} \text{(iv)}\;& B - A = \{4,8,16,20\} \end{aligned} \)

\( \begin{aligned} \text{(v)}\;& C - A = \{2,4,8,10,14,16\} \end{aligned} \)

\( \begin{aligned} \text{(vi)}\;& D - A = \{5,10,20\} \end{aligned} \)

\( \begin{aligned} \text{(vii)}\;& B - C = \{20\} \end{aligned} \)

\( \begin{aligned} \text{(viii)}\;& B - D = \{4,8,12,16\} \end{aligned} \)

\( \begin{aligned} \text{(ix)}\;& C - B = \{2,6,10,14\} \end{aligned} \)

\( \begin{aligned} \text{(x)}\;& D - B = \{5,10,15\} \end{aligned} \)

\( \begin{aligned} \text{(xi)}\;& C - D = \{2,4,6,8,12,14,16\} \end{aligned} \)

\( \begin{aligned} \text{(xii)}\;& D - C = \{5,15,20\} \end{aligned} \)

Q10. If X= { a, b, c, d } and Y = { f, b, d, g}, find
(i) X – Y
(ii) Y – X
(iii) X ∩ Y

Solution

Solution: The difference of two sets consists of elements belonging to the first set but not to the second, while the intersection consists of elements common to both sets.

\( \begin{aligned} X &= \{a, b, c, d\} \\ Y &= \{f, b, d, g\} \end{aligned} \)

\( \begin{aligned} \text{(i)}\;& X - Y = \{a, c\} \end{aligned} \)

\( \begin{aligned} \text{(ii)}\;& Y - X = \{f, g\} \end{aligned} \)

\( \begin{aligned} \text{(iii)}\;& X \cap Y = \{b, d\} \end{aligned} \)

Q11. If R is the set of real numbers and Q is the set of rational numbers, then what is R – Q?

Solution

Solution: Let \(R\) denote the set of all real numbers and \(Q\) denote the set of all rational numbers. Every real number is either rational or irrational.

\( \begin{aligned} R &= \{\text{all real numbers}\} \\ Q &= \{\text{all rational numbers}\} \\ R - Q &= \{\text{real numbers which are not rational}\}. \end{aligned} \)

Hence, \(R - Q\) represents the set of all irrational numbers.

Q12. State whether each of the following statement is true or false. Justify your answer.
(i) { 2, 3, 4, 5 } and { 3, 6} are disjoint sets.
(ii) { a, e, i, o, u } and { a, b, c, d }are disjoint sets.
(iii) { 2, 6, 10, 14 } and { 3, 7, 11, 15} are disjoint sets.
(iv) { 2, 6, 10 } and { 3, 7, 11} are disjoint sets.

Solution

Solution: Two sets are said to be disjoint if they have no element in common, that is, if their intersection is the empty set.

\( \begin{aligned} \text{(i)}\;& \{2,3,4,5\} \text{ and } \{3,6\}. \\ &\text{The element } 3 \text{ is common to both sets.} \\ &\Rightarrow \text{The statement is false.} \end{aligned} \)

\( \begin{aligned} \text{(ii)}\;& \{a,e,i,o,u\} \text{ and } \{a,b,c,d\}. \\ &\text{The element } a \text{ is common to both sets.} \\ &\Rightarrow \text{The statement is false.} \end{aligned} \)

\( \begin{aligned} \text{(iii)}\;& \{2,6,10,14\} \text{ and } \{3,7,11,15\}. \\ &\text{No element of the first set appears in the second set.} \\ &\Rightarrow \text{The statement is true.} \end{aligned} \)

\( \begin{aligned} \text{(iv)}\;& \{2,6,10\} \text{ and } \{3,7,11\}. \\ &\text{These two sets also have no element in common.} \\ &\Rightarrow \text{The statement is true.} \end{aligned} \)