Ordered pair, Cartesian product of two sets - Practice Questions & MCQ

Ordered pair

A pair of elements grouped together in a particular order is known as an ordered pair.
e.g. : $(a, b),(3,5),(1,0) \ldots$

The ordered pairs ( $\mathrm{a}, \mathrm{b}$ ) and ( $\mathrm{b}, \mathrm{a})$ are different.
Two ordered pairs are equal, if and only if the corresponding first elements are equal and the second elements are also equal.
i.e. $(x, y)=(u, v)$ if and only if $x=u, y=v$.

Cartesian product
The cartesian product of two nonempty sets $A$ and $B$ is the set of all ordered pairs $(x, y)$, where $x \in A$ and $y \in B$.
Symbolically, we write it as $\mathrm{A} \times \mathrm{B}$ and it is read as ' A cross B '.

$
A \times B=\{(a, b): a \in A, b \in B\}
$
For example, If $A=\{1,2\}$ and $B=\{a, b\}$
Then $A \times B=\{(1, a),(1, b),(2, a),(2, b)\}$
Note
m$A \times A \times A=\{(a, b, c): a, b, c \in A\}$. Here $(a, b, c)$ is called an ordered triplet.
$R \times R=\{(x, y): x, y \in R\}$ and $R \times R \times R=(x, y, z): x, y, z \in R\}$

Number of elements in A x B

If there are $p$ elements in $A$ and $q$ elements in $B$, then there will be $p q$ elements in $A \times B$, i.e., if $n(A)=p$ and $n(B)=q$, then $n(A \times B)=p q$.

If $A, B, C$ and $D$ are any four sets, then

$
\begin{aligned}
& A X(B \cup C)=(A \times B) \cup(A \times C) \\
& A X(B \cap C)=(A \times B) \cap(A \times C) \\
& A X(B-C)=(A \times B)-(A \times C)
\end{aligned}
$
$
\begin{aligned}
& (A \times B) \cap(C \times D)=(A \cap C) \times(B \times D) \\
& \text { If } A \subseteq B \text {, then }(A \times C) \subseteq(B \times C)
\end{aligned}
$
If $A \subseteq B$, then $A X A \subseteq(A X B) \cap(B \times A)$ If $A \subseteq B$ and $C \subseteq D$, then $A X C \subseteq B \times D$