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 $y$.

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 $y$.

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 $a+b=5$.

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 .

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 $B \times A$.

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.

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 $(A \times B) \cap (B \times A)$ ?

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 $n[ (A \times B) \cap (B \times A)]$.

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.

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.

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.

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 $A \times A \times A$.

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 $A \times (B \cap \phi ) = \phi$

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 $A \times A \times A$.

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) $B \times B$

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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