`star` Introduction

`star` Cartesian Products of Sets

`star` Cartesian Products of Sets

`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.

`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.

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.

`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.`

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?

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)`

(i) `A × (B ∩ C)` (ii) `(A × B) ∩ (A × C)`

(iii) `A × (B ∪ C)` (iv) `(A × B) ∪ (A × C)`

(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.`

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?

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.

A = set of first elements `= {p, m}`

B = set of second elements `= {q, r}.`