Mathematics Cartesian Products of Sets
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### ✎Topics Covered

star Introduction
star Cartesian Products of Sets

### Introduction

color{green} ✍️  In our daily life, we come across many patterns that characterise relations such as brother and sister, father and son, teacher and student.

color{green} ✍️  In mathematics also, we come across many relations such as number m is less than number n, line l is parallel to line m, set A is a subset of set B.

color{green} ✍️  In all these, we notice that a relation involves pairs of objects in certain order.

color{green} ✍️  In this Chapter, we will learn how to link pairs of objects from two sets and then introduce relations between the two objects in the pair.

color{green} ✍️  Finally, we will learn about special relations which will qualify to be functions.

color{green} ✍️  The concept of function is very important in mathematics since it captures the idea of a mathematically precise correspondence between one quantity with the other.

### Cartesian Products of Sets

An color{blue}ul("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 ∈ P and q ∈ Q .

color{purple}ul(✓✓) color{purple} mathbf(" DEFINITION ALERT")
Given two non-empty sets P and Q. The cartesian product P × Q is the set of all ordered pairs of elements from P and Q, i.e.,

 \ \ \ \ \ \ color{green} (P × Q = { (p,q) : p ∈ P, q ∈ Q })

If either P or Q is the null set, then P × Q will also be empty set, i.e., P × Q = φ

color{red}☛ color{red}(" Example") : Let P={1} , Q={2} qquad qquad P xx Q = { (1,2)}

P={2}, Q= {1} qquad qquad Q xx P={(2,1)}

P xx Q ne Q xx P

In most of the cases P xx Q ne Q xx P

color{green} ✍️  Suppose A is a set of 2 colours and B is a set of 3 objects, i.e.,

 \ \ \ \ \ \ \ \ \ color{blue } (A = {"red, blue"}) and color{blue } (B = {b, c, s},)
where b, c and s represent a particular bag, coat and shirt, respectively.

color{blue}( "How many pairs of coloured objects can be made from these two sets ?")

Proceeding in a very orderly manner, we can see that there will be 6 distinct pairs as given below:

 \ \ \ \ \ \ color{green} "(red, b), (red, c), (red, s), (blue, b), (blue, c), (blue, s)."
Thus, we get 6 distinct objects (Fig 1).

From the illustration given above we note that

 \ \ \ \ \ \ \ \ \ color{blue } (A × B = {"(red,b), (red,c), (red,s), (blue,b), (blue,c), (blue,s)"}.)

 "Again, consider the two sets :"

color{purple}( ✍️ A = {DL, MP, KA},)  where DL, MP, KA represent Delhi, Madhya Pradesh and Karnataka, respectively and color{purple} (B = {01,02, 03})  representing codes for the licence plates of vehicles issued by DL, MP and KA .

● If the three states, Delhi, Madhya Pradesh and Karnataka were making codes for the licence plates of vehicles, with the restriction that the code begins with an element from set A,

● which are the pairs available from these sets and how many such pairs will there be (Fig 2)?

● The available pairs are:

(DL,01), (DL,02), (DL,03), (MP,01), (MP,02), (MP,03), (KA,01), (KA,02), (KA,03)

and the product of set A and set B is given by

 color{purple}(A × B = {(DL,01), (DL,02), (DL,03), (MP,01), (MP,02), (MP,03), (KA,01), (KA,02), (KA,03)}.)

● It can easily be seen that there will be 9 such pairs in the Cartesian product, since there are 3 elements in each of the sets A and B. This gives us 9 possible codes.

\color{red} \ox \color{red} \mathbf(COMMON \ CONFUSION)  Also note that the order in which these elements are paired is crucial.
For example, the code (DL, 01) will not be the same as the code (01, DL).

color{green} ✍️ color{green} mathbf("KEY POINTS")
(i) Two ordered pairs are equal, if and only if the corresponding first elements are equal and the second elements are also equal.

(ii) If there are p elements in A and q elements in B, then there will be pq elements in A × B, i.e., if color{fuchsia} [n(A) = p] and color{fuchsia} [n(B) = q,] then color{fuchsia} [n(A × B) = pq.]

(iii) If A and B are non-empty sets and either A or B is an infinite set, then so is A × B.

(iv) \ \ A × A × A = {(a, b, c) : a, b, c ∈ A}. Here (a, b, c) is called an ordered triplet.
Q 3007380288

If (x + 1, y – 2) = (3,1), find the values of x and y.

Solution:

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

Therefore x + 1 = 3 and y – 2 = 1.
Solving we get x = 2 and y = 3.
Q 3087480387

If P = {a, b, c} and Q = {r}, form the sets P × Q and Q × P. Are these two products equal?

Solution:

By the definition of the cartesian product,
P × Q = {(a, r), (b, r), (c, r)} and Q × P = {(r, a), (r, b), (r, c)}
Since, by the definition of equality of ordered pairs, the pair (a, r) is not equal to the pair
(r, a), we conclude that P × Q ≠ Q × P.
However, the number of elements in each set will be the same.
Q 3067580485

Let A = {1,2,3}, B = {3,4} and C = {4,5,6}. Find
(i) A × (B ∩ C) (ii) (A × B) ∩ (A × C)
(iii) A × (B ∪ C) (iv) (A × B) ∪ (A × C)

Solution:

(i) By the definition of the intersection of two sets, (B ∩ C) = {4}.
Therefore, A × (B ∩ C) = {(1,4), (2,4), (3,4)}.
(ii) Now (A × B) = {(1,3), (1,4), (2,3), (2,4), (3,3), (3,4)} and (A × C) = {(1,4), (1,5), (1,6), (2,4), (2,5), (2,6), (3,4), (3,5), (3,6)}
Therefore, (A × B) ∩ (A × C) = {(1, 4), (2, 4), (3, 4)}.
(iii) Since, (B ∪ C) = {3, 4, 5, 6}, we have A × (B ∪ C) = {(1,3), (1,4), (1,5), (1,6), (2,3), (2,4), (2,5), (2,6), (3,3), (3,4), (3,5), (3,6)}.
(iv) Using the sets A × B and A × C from part (ii) above, we obtain (A × B) ∪ (A × C) = {(1,3), (1,4), (1,5), (1,6), (2,3), (2,4), (2,5), (2,6),
(3,3), (3,4), (3,5), (3,6)}.
Q 3007780688

If P = {1, 2}, form the set P × P × P.

Solution:

SolutionWe have, P × P × P = {(1,1,1), (1,1,2), (1,2,1), (1,2,2), (2,1,1), (2,1,2), (2,2,1), (2,2,2)}.
Q 3037880782

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

Solution:

The Cartesian product R × R represents the set R × R={(x, y) : x, y ∈ R} which represents the coordinates of all the points in two dimensional space and the cartisian product R × R × R represents the set R × R × R ={(x, y, z) : x, y, z ∈ R} which represents the coordinates of all the points in three-dimensional space.
Q 3047880783

If A × B ={(p, q),(p, r), (m, q), (m, r)}, find A and B.

Solution:

A = set of first elements = {p, m}
B = set of second elements = {q, r}.