$A \times B = \left\{ {\left( {a,b} \right):a \in A,b \in B} \right\}$

If either $A$ or $B$ is the null set, then $A \times B$ will also be empty set, i.e., $A \times B =\phi $

An ordered pair of elements taken from any two sets P and Q is a pair of elements written in small brackets and grouped together in a particular order, i.e., (p,q), $p \in P$ and $q \in Q$ .

Ordered pair (3,2) is equal to ordered pair (3,2)

Ordered pair (2,3) is not equal to ordered pair (3,2)

Ordered pair (1,1) is equal to ordered pair (1,1)

- If there are $p$ elements in $A$ and $q$ elements in $B$, then there will be $pq$ elements in $A \times B$, i.e., if $n\left( A \right) = p$ and $n\left( B \right) = q$, then $n\left( {A \times B} \right) = pq$.
- If there are $p$ elements in $A$ and $q$ elements in $B$ and $r$ in $C$, then there will be $pqr$ elements in $A \times B \times C$, i.e., if $n\left( A \right) = p$ and $n\left( B \right) = q$, then $n\left( {A \times B} \right) = pq$.
- If $A$ and $B$ are non-empty sets and either $A$ or $B$ is an infinite set , then so is $A \times B$.

A={1,2} , B={} , then $A \times B =\phi $

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

- $A \times A \times A = \left\{ {\left( {a,b,c} \right):a,b,c \in A} \right\}$.
Here ${\left( {a,b,c} \right)}$ is called an
*ordered triplet* - In general if $A_1,A_2,A_3....A_n$ are any n sets,then Cartesian product of $A_1,A_2,A_3....A_n$ and is denoted by $A_1 \times A_2 \times A_3 ....\times A_n$ and it is defined as

$A_1 \times A_2 \times A_3 ....\times A_n =\left\{(a_1 ,a_2 ,...,a_n) : a_i \in A_i , 1 \leq i\leq n \right\}$ and $(a_1 ,a_2 ,...,a_n)$ is called the ordered n-tuple

- Suppose we need to find the Cartesian Product of the sets A={1,2,3} and B={a,b,c}
- Take first element of First set and then form all the ordered pair with the all the element of the second set one by one

- Ordered pair obtained from this are { (1,a),(1,b),(1,c)}
- Now take the second element of first set and then form all the ordered pair with the all the element of the second set one by one

- Ordered pair obtained from this are { (2,a),(2,b),(2,c)}
- Now take the third element(last element) of first set and then form all the ordered pair with the all the element of the second set one by one

- Ordered pair obtained from this are { (3,a),(3,b),(3,c)}
- So, total ordered pair obtained are {(1,a),(1,b),(1,c),(2,a),(2,b),(2,c),(3,a),(3,b),(3,c)} and This is cartesian product of $A \times B$
- we can also represent this cartesian product as below

$P = \left\{ {1,2} \right\}$ and $Q = \left\{ {5,4,2} \right\}$

We know that the Cartesian product $P \times Q$ of two non-empty sets $P$ and $Q$ is defined as

$P \times Q = \left\{ {\left( {p,q} \right):p \in P,q \in Q} \right\}$

Therefore,

$P \times Q = \left\{ {\left( {1,5} \right),\left( {1,4} \right),\left( {1,2} \right),\left( {2,5} \right),\left( {2,4} \right),\left( {2,2} \right)} \right\}$

$Q \times P = \left\{ {\left( {5,1} \right),\left( {5,2} \right),\left( {4,1} \right),\left( {4,2} \right),\left( {2,1} \right),\left( {2,2} \right)} \right\}$

It is known that for any non-empty set $A$, $A \times A \times A$ is defined as

$A \times A \times A = \left\{ {\left( {a,b,c} \right):a,b,c \in A} \right\}$

It is given that $A = \left\{ { - 1,1} \right\}$

Therefore,

$A \times A \times A\ =\left\{ \left( { - 1, - 1, - 1} \right),\left( { - 1, - 1,1} \right), \left( { - 1,1, - 1} \right),\left( { - 1,1,1} \right) ,\left( {1, - 1, - 1} \right),\left( {1, - 1,1} \right),\left( {1,1, - 1} \right) \right\}$

Since the ordered pairs are equal, the corresponding elements are equal.

So, p + 11 = 2 and q +5 = 1.

Solving we get p = -9 and q = -4.

If R is the set of all real numbers, what do the cartesian products $R \times R$ and $R \times R \times R$ represent?

The Cartesian product $R \times R$ represents the set

$R \times R={(x, y) : x, y \in R}$

which represents the coordinates of all the points in two dimensional space and the

The Cartesian product $R \times R \times R$ represents the set set

$R \times R \times R ={(x, y, z) : x, y, z \in R}$

which represents the coordinates of all the points in three-dimensional space.

ii.$ A \times (B \cap C) = (A \times B) \cap (A \times C)$

iii. $ A \times (B - C) = (A \times B) - (A \times C)$

iv. $ (A \times B) \cap ( C \times D) =(A \cap C) \times (B \cap D)$

v. if $ A \subset B$ then $ A \times C \subset B \times C$

vi. $ A \times B \ne B \times A$

vii. $ (A \times B) \times C \ne A \times (B \times C)$

**Notes****Assignments**