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

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

- Two ordered pairs are equal, if and only if the corresponding first elements
are equal and the second elements are also equal.

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

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

- 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 $A$ and $B$ are non-empty sets and either $A$ or $B$ is an infinite set, then so is $A \times B$.
__Cartesian product of three sets__

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

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

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$

- Cartesian Products
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- What is relations?
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- What is Function
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- Domain of Function
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- Range of Function
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- Identity Function
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- Constant Function
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- Linear Function
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- Modules Function
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- Greatest Integer Function
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- Polynomial Function
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- Algebra of Real Function

Class 11 Maths Class 11 Physics