\displaystyle \textbf{Question 1: }\text{(i) If }\left(\frac{a}{3}+1,\;b-\frac{2}{3}\right)=\left(\frac{5}{3},\;\frac{1}{3}\right),
\displaystyle \text{find the values of }a\text{ and }b.\text{ (ii) If }(x+1,1)=(3,y-2),\text{ find }x\text{ and }y.
\displaystyle \text{Answer:}
\displaystyle \text{(i) }\left(\frac{a}{3}+1,\;b-\frac{2}{3}\right)=\left(\frac{5}{3},\;\frac{1}{3}\right)
\displaystyle \text{By the definition of equality of ordered pairs,}
\displaystyle \frac{a}{3}+1=\frac{5}{3}\text{ and }b-\frac{2}{3}=\frac{1}{3}
\displaystyle \Rightarrow \frac{a}{3}=\frac{2}{3}\text{ and }b=1
\displaystyle \Rightarrow a=2\text{ and }b=1
\displaystyle \text{(ii) }(x+1,1)=(3,y-2)
\displaystyle \text{By the definition of equality of ordered pairs,}
\displaystyle x+1=3\text{ and }y-2=1
\displaystyle \Rightarrow x=2\text{ and }y=3
\displaystyle \\

\displaystyle \textbf{Question 2: }\text{If the ordered pairs }(x,-1)\text{ and }(5,y)\text{ belong to the set}
\displaystyle \{(a,b):b=2a-3\},\text{ find the values of }x\text{ and }y.
\displaystyle \text{Answer:}
\displaystyle (x,-1)\in\{(a,b):b=2a-3\}
\displaystyle \Rightarrow a=x,\;b=-1\text{ and }-1=2x-3
\displaystyle \Rightarrow 2x=2
\displaystyle \Rightarrow x=1
\displaystyle (5,y)\in\{(a,b):b=2a-3\}
\displaystyle \Rightarrow a=5,\;b=y
\displaystyle \Rightarrow y=2(5)-3=7
\displaystyle \therefore x=1\text{ and }y=7
\displaystyle \\

\displaystyle \textbf{Question 3: }\text{If }a\in\{1,2,3,4,5\}\text{ and }b\in\{0,3,6\},\text{ write the set of}
\displaystyle \text{all ordered pairs }(a,b)\text{ such that }a+b=5.
\displaystyle \text{Answer:}
\displaystyle \text{Given }a\in\{1,2,3,4,5\}\text{ and }b\in\{0,3,6\}.
\displaystyle \text{Checking the possible values of }b:
\displaystyle b=6\Rightarrow a=-1,\text{ which is not in the given set}
\displaystyle b=3\Rightarrow a=2
\displaystyle b=0\Rightarrow a=5
\displaystyle \therefore \text{The required set is }\{(2,3),(5,0)\}.
\displaystyle \\

\displaystyle \textbf{Question 4: }\text{If }a\in\{2,4,6,9\}\text{ and }b\in\{4,6,18,27\},\text{ form the set of}
\displaystyle \text{all ordered pairs }(a,b)\text{ such that }a\text{ divides }b\text{ and }a<b.
\displaystyle \text{Answer:}
\displaystyle \text{Given }a\in\{2,4,6,9\}\text{ and }b\in\{4,6,18,27\}.
\displaystyle 2\text{ divides }4,6,18\text{ and }2<4,6,18.
\displaystyle 4\text{ divides only }4,\text{ but }4\nless4.
\displaystyle 6\text{ divides }18\text{ and }6<18.
\displaystyle 9\text{ divides }27\text{ and }9<27.
\displaystyle \therefore \text{The required set is }
\displaystyle \{(2,4),(2,6),(2,18),(6,18),(9,27)\}.
\displaystyle \\

\displaystyle \textbf{Question 5: }\text{If }A=\{1,2\},\;B=\{1,3\},\text{ find }A\times B\text{ and }B\times A.
\displaystyle \text{Answer:}
\displaystyle \text{Given }A=\{1,2\}\text{ and }B=\{1,3\}.
\displaystyle A\times B=\{(1,1),(1,3),(2,1),(2,3)\}.
\displaystyle B\times A=\{(1,1),(1,2),(3,1),(3,2)\}.
\displaystyle \\

\displaystyle \textbf{Question 6: }\text{Let }A=\{1,2,3\}\text{ and }B=\{3,4\}.\text{ Find }A\times B\text{ and show it}
\displaystyle \text{graphically.}
\displaystyle \text{Answer:}
\displaystyle \text{Given }A=\{1,2,3\}\text{ and }B=\{3,4\}.
\displaystyle A\times B=\{(1,3),(1,4),(2,3),(2,4),(3,3),(3,4)\}.
\displaystyle \text{Graphically, plot the points }(1,3),(1,4),(2,3),(2,4),(3,3)\text{ and }(3,4)\text{ on the Cartesian plane.}
\displaystyle \\

\displaystyle \textbf{Question 7: }\text{If }A=\{1,2,3\}\text{ and }B=\{2,4\},\text{ find }A\times B,\;B\times A,\;A\times A,
\displaystyle B\times B\text{ and }(A\times B)\cap(B\times A).
\displaystyle \text{Answer:}
\displaystyle \text{Given }A=\{1,2,3\}\text{ and }B=\{2,4\}.
\displaystyle A\times B=\{(1,2),(1,4),(2,2),(2,4),(3,2),(3,4)\}.
\displaystyle B\times A=\{(2,1),(2,2),(2,3),(4,1),(4,2),(4,3)\}.
\displaystyle A\times A=\{(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)\}.
\displaystyle B\times B=\{(2,2),(2,4),(4,2),(4,4)\}.
\displaystyle (A\times B)\cap(B\times A)=\{(2,2)\}.
\displaystyle \\

\displaystyle \textbf{Question 8: }\text{If }A\text{ and }B\text{ are two sets having }3\text{ elements in common,}
\displaystyle \text{and }n(A)=5,\;n(B)=4,\text{ find }n(A\times B)\text{ and }n[(A\times B)\cap(B\times A)].
\displaystyle \text{Answer:}
\displaystyle \text{Given }n(A)=5,\;n(B)=4\text{ and }n(A\cap B)=3.
\displaystyle n(A\times B)=n(A)\times n(B)=5\times4=20.
\displaystyle \text{Also, }(A\times B)\cap(B\times A)=(A\cap B)\times(A\cap B).
\displaystyle \therefore n[(A\times B)\cap(B\times A)]
\displaystyle =n(A\cap B)\times n(A\cap B)
\displaystyle =3\times3=9.
\displaystyle \therefore n(A\times B)=20\text{ and }n[(A\times B)\cap(B\times A)]=9.
\displaystyle \\

\displaystyle \textbf{Question 9: }\text{Let }A\text{ and }B\text{ be two sets. Show that }A\times B\text{ and }B\times A
\displaystyle \text{have an element in common iff }A\text{ and }B\text{ have an element in common.}
\displaystyle \text{Answer:}
\displaystyle \text{Suppose }(x,y)\in(A\times B)\cap(B\times A).
\displaystyle \text{Then }(x,y)\in A\times B\text{ and }(x,y)\in B\times A.
\displaystyle \Rightarrow x\in A,\;y\in B\text{ and }x\in B,\;y\in A.
\displaystyle \Rightarrow x\in A\cap B\text{ and }y\in A\cap B.
\displaystyle \therefore A\cap B\neq\emptyset.
\displaystyle \text{Conversely, let }a\in A\cap B.
\displaystyle \Rightarrow a\in A\text{ and }a\in B.
\displaystyle \therefore (a,a)\in A\times B\text{ and }(a,a)\in B\times A.
\displaystyle \Rightarrow (a,a)\in(A\times B)\cap(B\times A).
\displaystyle \therefore A\times B\text{ and }B\times A\text{ have a common element.}
\displaystyle \text{Hence, }A\times B\text{ and }B\times A\text{ have an element in common}
\displaystyle \text{if and only if }A\text{ and }B\text{ have an element in common.}
\displaystyle \\

\displaystyle \textbf{Question 10: }\text{Let }A\text{ and }B\text{ be two sets such that }n(A)=3\text{ and }n(B)=2.
\displaystyle \text{If }(x,1),(y,2),(z,1)\in A\times B,\text{ find }A\text{ and }B,\text{ where }x,y\text{ and }z
\displaystyle \text{are distinct elements.}
\displaystyle \text{Answer:}
\displaystyle \text{Since }(x,1),(y,2),(z,1)\in A\times B,
\displaystyle x,y,z\in A\text{ and }1,2\in B.
\displaystyle \text{Given }n(A)=3\text{ and }x,y,z\text{ are distinct,}
\displaystyle \therefore A=\{x,y,z\}.
\displaystyle \text{Also }n(B)=2\text{ and }1,2\in B.
\displaystyle \therefore B=\{1,2\}.
\displaystyle \\

\displaystyle \textbf{Question 11: }\text{Let }A=\{1,2,3,4\}\text{ and }R=\{(a,b):a\in A,\;b\in A,\;a\text{ divides }b\}.
\displaystyle \text{Write }R\text{ explicitly.}
\displaystyle \text{Answer:}
\displaystyle \text{Given }A=\{1,2,3,4\}.
\displaystyle 1\text{ divides }1,2,3,4.
\displaystyle 2\text{ divides }2,4.
\displaystyle 3\text{ divides }3.
\displaystyle 4\text{ divides }4.
\displaystyle \therefore R=\{(1,1),(1,2),(1,3),(1,4),(2,2),(2,4),(3,3),(4,4)\}.
\displaystyle \\

\displaystyle \textbf{Question 12: }\text{If }A=\{-1,1\},\text{ find }A\times A\times A.
\displaystyle \text{Answer:}
\displaystyle \text{Given }A=\{-1,1\}.
\displaystyle A\times A=\{(-1,-1),(-1,1),(1,-1),(1,1)\}.
\displaystyle \therefore A\times A\times A=
\displaystyle \{(-1,-1,-1),(-1,-1,1),(-1,1,-1),(-1,1,1),
\displaystyle (1,-1,-1),(1,-1,1),(1,1,-1),(1,1,1)\}.
\displaystyle \\

\displaystyle \textbf{Question 13: }\text{State whether each of the following statements is true or false. If the}
\displaystyle \text{statement is false, rewrite it correctly.}
\displaystyle \text{(i) If }P=\{m,n\}\text{ and }Q=\{n,m\},\text{ then }P\times Q=\{(m,n),(n,m)\}.
\displaystyle \text{(ii) If }A\text{ and }B\text{ are non-empty sets, then }A\times B\text{ is a non-empty}
\displaystyle \text{set of ordered pairs }(x,y)\text{ such that }x\in B\text{ and }y\in A.
\displaystyle \text{(iii) If }A=\{1,2\},\;B=\{3,4\},\text{ then }A\times(B\cap\phi)=\phi.
\displaystyle \text{Answer:}
\displaystyle \text{(i) False.}
\displaystyle P\times Q=\{(m,m),(m,n),(n,m),(n,n)\}.
\displaystyle \text{(ii) False.}
\displaystyle \text{If }A\text{ and }B\text{ are non-empty sets, then }A\times B\text{ is a non-empty set}
\displaystyle \text{of ordered pairs }(x,y)\text{ such that }x\in A\text{ and }y\in B.
\displaystyle \text{(iii) True, since }B\cap\phi=\phi\text{ and }A\times\phi=\phi.
\displaystyle \\

\displaystyle \textbf{Question 14: }\text{If }A=\{1,2\},\text{ form the set }A\times A\times A.
\displaystyle \text{Answer:}
\displaystyle \text{Given }A=\{1,2\}.
\displaystyle A\times A=\{(1,1),(1,2),(2,1),(2,2)\}.
\displaystyle \therefore A\times A\times A=(A\times A)\times A
\displaystyle =\{(1,1,1),(1,1,2),(1,2,1),(1,2,2),
\displaystyle (2,1,1),(2,1,2),(2,2,1),(2,2,2)\}.
\displaystyle \\

\displaystyle \textbf{Question 15: }\text{If }A=\{1,2,4\}\text{ and }B=\{1,2,3\},\text{ represent the following}
\displaystyle \text{sets graphically: (i) }A\times B\qquad\text{(ii) }B\times A\qquad\text{(iii) }A\times A\qquad\text{(iv) }B\times B.
\displaystyle \text{Answer:}
\displaystyle \text{Given }A=\{1,2,4\}\text{ and }B=\{1,2,3\}.
\displaystyle \text{(i) }A\times B=\{(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(4,1),(4,2),(4,3)\}.
\displaystyle \text{(ii) }B\times A=\{(1,1),(1,2),(1,4),(2,1),(2,2),(2,4),(3,1),(3,2),(3,4)\}.
\displaystyle \text{(iii) }A\times A=\{(1,1),(1,2),(1,4),(2,1),(2,2),(2,4),(4,1),(4,2),(4,4)\}.
\displaystyle \text{(iv) }B\times B=\{(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)\}.
\displaystyle \text{Graphically, plot the ordered pairs of each Cartesian product on the Cartesian plane.}
\displaystyle \\

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