Question 1: i) If  \Big( \frac{a}{3} + 1 , b - \frac{2}{3} \Big)  = \Big( \frac{5}{3} , \frac{1}{3} \Big) find the values of a and b (ii) if (x+1,1): (3,y -2) , find the values of x and y .

Answer:

i) \Big( \frac{a}{3} + 1 , b - \frac{2}{3} \Big)  = \Big( \frac{5}{3} , \frac{1}{3} \Big)

By definition of equality of ordered pairs, we have

\Rightarrow \frac{a}{3} +1 = \frac{5}{3}      and b - \frac{2}{3} =  \frac{1}{3}

\Rightarrow \frac{a}{3} = \frac{2}{3}     and b = 1

\Rightarrow a = 2       and    b= 1

ii) (x+1,1): (3,y -2)

By definition of equality of ordered pairs, we have

\Rightarrow x + 1 = 3      and y -2 = 1

\Rightarrow x = 2      and    y = 3

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Question 2: If the ordered pairs (x,-1) and (5,y) belong to the set \{ (a,b):b=2a-3 \} , find the values of x and y .

Answer:

The ordered pairs (x,-1) and (5,y) belong to the set \{ (a,b):b=2a-3 \}

Thus we have

x = a and -1 = b such that b = 2a - 3

\therefore -1 = 2x - 3

\Rightarrow 2x = 2 \Rightarrow x = 1

Also 5-a and y - b such that b - 2a = 3

\Rightarrow y = 2(5) - 3 = 10 - 3 = 7

Thus we get x = 1 and y = 7

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Question 3: If a \in \{ 1,2,3,4,5\} and b \in \{ 0,3,6 \} , write the set of all ordered pairs (a,b) such that a+b=5 .

Answer:

Given a \in \{ 1,2,3,4,5\} and b \in \{ 0,3,6 \}

Since a + b = 5 , we know that the possible ordered pairs are \{ (-1, 6), (2, 3), (5, 0) \} 

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Question 4: If a \in \{ 2, 4,6,9 \} and b \in \{4,6,18,27 \} , then form the set of all ordered pairs (a,b) such that a divides b and a <b .

Answer:

Given a \in \{ 2, 4,6,9 \} and b \in \{4,6,18,27 \} . Here

2 divides 4, 6, 18 and 2 is less than all of them

6 divides  18 and 6 is less than 18

9 divides 18, 27 and 9 is less than 18 and 27

Therefore set of all ordered pairs (a, b) such that a divides b and a < b Hence R = \{ (2, 4), (2, 6), (2, 18), (6, 18), (9, 18), (9, 27)  \}

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Question 5: If A= \{ 1, 2 \} B=\{ 1,3 \} , find A \times B and B \times A .

Answer:

Given A= \{ 1, 2 \} B=\{ 1,3 \}

\therefore A \times B = \{ 1, 2 \} \times \{ 1,3 \} = \{  (1, 1), (1, 3), (2, 1), (2, 2) \} 

B \times A = \{ 1,3 \} \times \{ 1, 2 \} = \{ (1, 1), (1, 2), (3, 1), (3, 2) \}

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Question 6: Let A = \{1, 2, 3 \} and B = \{ 3, 4 \} . Find A \times B and show it graphically.

Answer:

Given A = \{1, 2, 3 \} and B = \{ 3, 4 \}

A \times B = \{1, 2, 3 \} \times \{ 3, 4 \} = \{ (1, 3), (1, 4),  (2, 3), (2, 4),  (3, 3), (3, 4) \}

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Question 7: If A = \{1, 2, 3\} and B = \{2,4\} , what are A \times B, B \times A, A \times A, B \times B , and (A \times B) \cap (B \times A) ?

Answer:

Given A = \{1, 2, 3\} and B = \{2,4\}

\therefore A \times B = \{1, 2, 3\} \times  \{2,4\} = \{ (1, 2), (1, 4), (2, 2), (2, 4), (3, 2), (3, 4) \}

B \times A = \{2,4\} \times \{1, 2, 3\}   = \{ (2, 1), (2, 2), (2, 3), (4, 1), (4, 2), (4, 3)  \}

A \times A = \{1, 2, 3\} \times \{1, 2, 3\}  \\ \hspace*{1.2cm} = \{ (1, 1), (1,2), (1, 3), (2, 1), (2,2), (2, 3),(3, 1), (3,2), (3, 3) \}

\therefore  (A \times B) \cap (B \times A) = \{ (2, 2) \} 

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Question 8: If A and B are two sets haying 3 elements in common. It n(A)=5, n(B)=4 , find n(A \times B) and n[ (A \times B) \cap (B \times A)] .

Answer:

Given n(A) = 5 and n(B ) = 4

Therefore n(A \times B) = 5 \times 4 = 20

A and B are two sets having 3 elements in common

Let A = \{ a, a, a, b, c \} and B = \{ a, a, a, d \}

\therefore A \times B = \{ (a, a), (a, a), (a, a), (a, d), (a, a), (a, a), (a, a), \\  \hspace*{2.4cm} (a, d), (a, a), (a, a), (a, a), (a, d), (b, a), (b, a), \\ \hspace*{2.4cm} (b, a), (b, d), (c, a), (c, a), (c, a), (c, d) \}

B \times A = \{  (a, a),  (a, a),  (a, a),  (a, b),  (a, c), (a, a),  (a, a),  (a, a),  \\ \hspace*{2.1cm} (a, b),  (a, c), (a, a),  (a, a),  (a, a),  (a, b),  (a, c), \\ \hspace*{2.1cm} (d, a),  (d, a),  (d, a),  (d, b),  (d, c) \}

(A \times B) \cap (B \times A) = \{ (a, a), (a, a), (a, a), (a, a), (a, a), (a, a), (a, a), (a, a), (a, a) \}

\therefore n [A \times B \cap B \times A] = 9

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Question 9: Let A and B be two sets. Show that the sets A \times B and B \times A have an element in common iff the sets A and B have an element in common.

Answer:

Case 1: Let A = \{ a, b, c \} and B = \{ e, f \}

\therefore A \times B = \{ a, b, c \}  \times \{ e, f \} = \{ (a, e), (a, f), (b, e), (b, f), (c, e), (c, f) \}

B \times A = \{ e, f \} \times \{ a, b, c \} = \{ (e, a), (e, b), (e, c), (f, a), (f, b), (f, c) \}

Therefore they have no common elements.

Case 2: Let A = \{ a, b, c \} and B = \{ a, f \}

\therefore A \times B = \{ a, b, c \}  \times \{ a, f \} = \{ (a, a), (a, f), (b, a), (b, f), (c, a), (c, f) \}

B \times A = \{ a, f \} \times \{ a, b, c \} = \{ (a, a), (a, b), (a, c), (f, a), (f, b), (f, c) \}

Hence (A \times B) \cap (B \times A) = \{(a, a), (a, a) \}

Therefore A \times B and B \times A will have common elements if and only if sets A and set B have elements in common.

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Question 10: Let A and B be two sets such that n(A) = 3 and n(B) = 2 . If (x, 1), (y, 2), (z, 1) are in A \times B find A and B   where x, y , and z are distinct elements.

Answer:

Since (x, 1), (y, 2), (z, 1) are elements of A \times B , therefore x, y, z, \in A and 1, 2, \in B

It is given that n(A) = 3 and n(B) = 3

\Rightarrow A = \{ x, y, z \}

Similarly, 1, 2 \in B and n(B) = 2

\Rightarrow B = \{1, 2 \}

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Question 11: Let A = \{1, 2, 3, 4 \} and R = \{(a, b) : a \in A, b \in A, a divides b \} . Write R explicitly.

Answer:

Given A = \{1, 2, 3, 4 \}

R = \{(a, b) : a \in A, b \in A, a divides b \}

We know: 1 divides 1, 2, 3, and 4

2 divides 2 and 4

3 divides 3

4 divides 4

\therefore  R = \{ (1, 1,), (1, 2), (1, 3), (1,4), (2,2), (2,4), (3,3), (4,4) \}

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Question 12: If A= \{-1,1 \} , find A \times A \times A .

Answer:

Given A= \{-1,1 \}

\therefore A \times A = \{-1,1 \} \times \{-1,1 \} = \{ (-1, -1), (-1, 1), (1, -1), (1, 1) \}

\therefore A \times A \times A = \{ (-1, -1, -1), (-1, -1, 1), (-1, 1, -1), (-1, 1, 1) , \\ \hspace*{3.0cm} (1, -1, -1), (1, -1, 1), (1, 1, -1), (1, 1, 1) \}

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Question 13: State whether each of the following statements are true or false. If the statement is false, re-write the given statement correctly:

(i) If P = \{m, n\} and Q = \{n, m\} , then P \times Q = \{ (m, n), (n, m) \}

(ii) If A and B are non-empty sets, then A \times B is a non-empty set of ordered pairs (x,y) such that x \in B  and y \in A .

(iii) If A = \{1, 2 \} , B = {3,4} , then A \times (B \cap \phi ) = \phi

Answer:

i) False

If P = \{m, n\} and Q = \{n, m\} , then

P \times Q = \{ (m, n), (m, m), (n,n) , (n, m) \}

ii) False

If A and B are non-empty sets then A \times B is a non-empty set of ordered pairs (x, y) such that x \in A and y \in B .

iii) True

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Question 14: If A = \{1, 2\} , form the set A \times A \times A .

Answer:

Given A = \{1, 2\}

A \times A = \{1, 2\} \times \{1, 2\} = \{ (1, 1), (1, 2), (2, 1), (2, 2) \} 

\therefore A \times A \times A = \{1, 2\} \times \{ (1, 1), (1, 2), (2, 1), (2, 2) \}  \\ \hspace*{2.0cm} = \{ (1, 1, 1), (1, 1, 2), (1, 2, 1), (1, 2, 2) , (2, 1, 1), (2, 1, 2), (2, 2, 1), (2, 2, 2) \}

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Question 15: If A = \{1, 2, 4 \} and B = \{1, 2, 3\} , represent following sets graphically: (i) A \times B (ii) B \times A (iii) A \times A  (iv) B \times B

Answer:

Given A = \{1, 2, 4 \} and B = \{1, 2, 3\}

i) A \times B = \{1, 2, 4 \} \times \{1, 2, 3\} \\ \hspace*{1.2cm} = \{ (1, 1), (1, 2), (1,3), (2, 1), (2, 2), (2,3),  (4, 1), (4, 2), (4,3) \}

ii)  B \times A = \{1, 2, 3\} \times \{1, 2, 4 \} \\ \hspace*{1.2cm} = \{ (1, 1), (1, 2), (1, 4), (2, 1), (2, 2), (2, 4),  (3, 1), (3, 2), (3, 4) \}

iii) A \times A =  \{1, 2, 4 \} \times \{1, 2, 4 \} \\ \hspace*{1.2cm} = \{ (1,1), (1,2), (1, 4), (2,1), (2,2), (2, 4),  (4,1), (4,2), (4, 4) \}

iv) B \times B =  \{1, 2, 3\} \times \{1, 2, 3\} \\ \hspace*{1.2cm} = \{ (1,1), (1,2), (1,3), (2,1), (2,2), (2,3),  (3,1), (3,2), (3,3) \}

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