Cartesian product of two sets

CARTESIAN PRODUCT
Janak singh Saud
saudjanaksingh@gmail.com
https://images.app.goo.gl/CjPHqUg6ZxC3ehER6
6/19/2020 JANAK SINGH SAUD 1

Assume the AASHIFA is only considering Car Company HONDA,
Car Company NISSAN, Car Company BMW and she is only
looking for white or red colour.
• Set A = {BMW, HONDA, NISSAN}
• Set B = {white , red}
The set of for this example as following:
https://images.app.goo.gl/TkXCbhW5RbZmCPvp7

BMW
HONDA
NISSAN
White
Red
A B
{(BMW, White), (BMW, Red), (HONDA, White), (HONDA, Red), (NISSAN, White), (Nissan, Red) }
The possible ordered pairs are :
Definition: Let A and B are two non- empty sets. Then the set of all possible ordered pairs in which the first element
from set A and second element from set B is called the Cartesian Product of Sets A and B. It is denoted by A × B which is
read as “A cross B”.

BMW
HONDA
NISSAN
White
Red
White
Red
White
Red
A B A ×B
(BMW, White)
(BMW, Red)
(HONDA, Red)
(HONDA, White)
(NISSAN, White)
(Nissan, Red)
A × B = {(BMW, White), (BMW, Red), (HONDA, White), (HONDA, Red), (NISSAN, White), (Nissan, Red) }
Tree Diagram

Table
B
A
× White Red
BMW
HONDA
NISSAN
A × B = {(BMW, White), (BMW, Red), (HONDA, White), (HONDA, Red), (NISSAN, White), (Nissan, Red) }
B
A
× White Red
BMW (BMW, White)
HONDA
NISSAN
B
A
× White Red
BMW (BMW, White) (BMW, Red)
HONDA
NISSAN
B
A
× White Red
HONDA (HONDA, White)
NISSAN
B
A
× White Red
HONDA (HONDA, White) (HONDA, Red)
NISSAN
B
A
× White Red
NISSAN (NISSAN, White)
B
A
× White Red
NISSAN (NISSAN, White) (NISSAN, Red)

If A = { a, b} and B = {1, 2, 3} , then find A ×B and B ×A
• Here A = {a, b} and B = {1, 2, 3}
• ∴ A ×B = {(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)}
• N(A) = 2 and N(B) = 3
• N(A×B) = 2×3 = 6
• B ×A = {(1, a), (1, b), (2, a), (2, b), (3, a), (3, b)}
• N(B×A) = 6
• NOTE:
1. A ×B ≠ B × A
2. A ×B = B × A if A and B are equal sets.
3. If N(A) = m and N(B) = n, then N(A×B) = m ×n
4. N(A×B) = N(B×A)
5. A ×B = ∅ if A or B is an empty set
a
b
1
2
3
A B
1
2
3
B A
a
b
6/19/2020
JANAK SINGH SAUD
7

EXAMPLE 2: IF A = {x∈N:x≤4}, find A×A
• Here A = {x∈N:x≤4}
A
A
×
1 2 3 4
1
2
3
4
∴ A = {1, 2, 3, 4}
A
A
×
1 2 3 4
1 (1, 1)
2
3
4
A
A
×
1 2 3 4
1 (1, 2)
2
3
4
A
A
×
1 2 3 4
1 (1, 3)
2
3
4
A
A
×
1 2 3 4
1 (1, 4)
2
3
4
A
A
×
1 2 3 4
1
2 (2, 1)
3
4
A
A
×
1 2 3 4
1
2 (2, 2)
3
4
A
A
×
1 2 3 4
1
2 (2, 3)
3
4
A
A
×
1 2 3 4
1
2 (2, 4)
3
4
A
A
×
1 2 3 4
1
2
3 (3, 1)
4
A
A
×
1
2
3 (3, 2)
4
A
A
×
(3, 3)
A
A
×
(3, 4)
A
A
×
(4, 1)
A
A
×
(4, 2)
A
A
×
(4, 3)
A
A
×
(4, 4)
Now, from the adjoining table;
A×A= {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4), (3, 1),
(3, 2), (3, 3), (3, 4), (4, 1), (4, 2), (4, 3), (4, 4)}

EXAMPLE 3: If A×B = {(2, 4), (2,5),(2,6),(3,4),(3,5),(3,6)}, then
find sets A and B. Also find n(A), n(B) and n(A×B)
Here, A×B = {(2, 4), (2,5),(2,6),(3,4),(3,5),(3,6)}
A = Set of first elements of the ordered pairs
= {2, 3}
B = Set of second elements of the ordered pairs
= {4, 5, 6 }
N(A) = 2 and N(B) = 3
∴N(A×B) = 2 ×3 = 6

If A = {x: x≤5, x∈N} and B = {x:x2- 4=0}, then find
A×B and B×A.
Here, A = {x: x≤5, x∈N}
= {1, 2, 3, 4, 5} .
And B = {x:x2- 4=0}
= {-2, 2}
Now, A ×B = {1, 2, 3, 4, 5} ×{- 2, 2}
= {(1, -2), (1, 2), (2, -2), (2, 2), (3, - 2), (3, 2), (4, -2),(4,2),(5,-
2),(4, 2)}
And B×A = {- 2, 2} × {1, 2, 3, 4, 5}
= {-2, 1),(-2,2),(-2,3),(-2,4),(-2,5),(2,1),(2,2),(2,3),(2,4),(2,5)}
∴x2 – 4 = 0
or, x2 – 22 = 0
or, (x+ 2) (x – 2) = 0
or, x + 2 = 0 , x – 2 = 0
or, x = - 2 and 2

1. In each of the following conditions, find
A×B
a) A = {2, 3} and B= {a, b}
b) A = {p, q} and B{1, 2}
c) A = {1, 2 3} and B = {r, s, t}
d) A = {2, 3, 4} and B = {2, 3, 4}
e) A = {1, 2, 3} and B = {0, 1, 2, 3}
f) A = {1, 2} and B = {4, 5, 6, 7}

2. Find A× B and B ×A in each of the
following cases:
a) P = {1, 2, 3} and Q = {a, b}
b) P = {2, 3, 4} and Q = {3, 4, 5}

3. SOLVE
• Given X = {1, 2, 3} and Y = {4, 5}. Show that X×Y ≠Y×X
• If M = {a, b, c} and N = {c, d}, then verify that M× N ≠ N × M

4. SOLVE
• If P× Q = {(1,m),(1,n),(2,m),(2,n),(3,m),(3,n)}, find P And Q
• If B×A = {(2, 5),(,3, 5), (4, 5),(2, 6), (3, 6),(4, 6)}, find A and B.

5. SOLVE
• If A = {x : x ≤3 and x ∈ 𝑁}, then find A ×A
• If P = {x : 7≤x ≤10}, then find P ×P.

6. SOLVE
• If P = {4, 5, 6} and Q = {1, 2}, show that P×Q in mapping diagram and
on graph
• If A = {x:x∈ 𝑁, 6< x <9}, find A ×A and show it in mapping diagram
and in lattice diagram.

H.W.- Cartesian Product
1. Find A× B and B ×A in each of the following cases if A = {1, 2, 3}
and B = {a, b} by tree diagram
2. If M = {a, b, c} and N = {c, d}, then verify that M× N ≠ N × M
3. If P× Q = {(1,m),(1,n),(2,m),(2,n),(3,m),(3,n)}, find P And Q
4. If P = {x : 7≤x ≤10}, then find P ×P.
5. If P = {4, 5, 6} and Q = {1, 2}, show that P×Q in mapping diagram
6. If A = {x:x∈ 𝑁, 6< x <9}, find A ×A and show it in Table Method and

Ram’s family- Revision of ordered pair
Dhanshyam 50 years
Gita 45 years
Ram 13 years
Hari 10 years
pramila 5 years
{(Dhanashyam, 50), (Gita, 45), (Ram, 13), (Hari, 10), (Pranila, 5)}
{(50, Dhanshyam), (45, Gita), (13, Ram), (10, Hari), (5, Pranila)}
Representing the above table in ordered pair form:

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