Sections Included:
1. Collection
2. Types of Collection
3. Sets
4. Commonly used Sets in Maths
5. Notation
6. Different Types of Sets
7. Venn Diagram
8. Operation on sets
9. Properties of Union of Sets
10. Properties of Intersection of Sets
11. Difference in Sets
12. Complement of Sets
13. Properties of Complement Sets
14. De Morgan’s Law
15. Inclusion Exclusion Principle
Sections Included:
1. Collection
2. Types of Collection
3. Sets
4. Commonly used Sets in Maths
5. Notation
6. Different Types of Sets
7. Venn Diagram
8. Operation on sets
9. Properties of Union of Sets
10. Properties of Intersection of Sets
11. Difference in Sets
12. Complement of Sets
13. Properties of Complement Sets
14. De Morgan’s Law
15. Inclusion Exclusion Principle
If you are looking for math video tutorials (with voice recording), you may download it on our YouTube Channel. Don't forget to SUBSCRIBE for you to get updated on our upcoming videos.
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Reference:
Nivera, G. C. (2013), Grade 7 Mathematics: Pattern and Practicalities. Don Bosco Press Inc. Makati City, Philippines.
In detail and In very simple method That can any one understand.
If you read this all you doubts about function will be clear.
because i have used very simple example and simple English words that you can pick quickly concept about functions.
#inshallah.
X std maths - Relations and functions (ex 1.1), Maths, IX std Maths, Samacheerkalvi maths, II year B.Ed., Pedagogy, Mathematics, Relation, Functions, Cartesian product, ordered pair, definition of cartesian product, standard infinite set, cartesian product of three sets,
If you are looking for math video tutorials (with voice recording), you may download it on our YouTube Channel. Don't forget to SUBSCRIBE for you to get updated on our upcoming videos.
https://tinyurl.com/y9muob6q
Also, please do visit our page, LIKE and FOLLOW us on Facebook!
https://tinyurl.com/ycjp8r7u
https://tinyurl.com/ybo27k2u
For more instructional resources, CLICK me here and DON'T FORGET TO SUBSCRIBE!
https://tinyurl.com/y9muob6q
LIKE and FOLLOW me here!
https://tinyurl.com/ycjp8r7u
https://tinyurl.com/ybo27k2u
Reference:
Nivera, G. C. (2013), Grade 7 Mathematics: Pattern and Practicalities. Don Bosco Press Inc. Makati City, Philippines.
In detail and In very simple method That can any one understand.
If you read this all you doubts about function will be clear.
because i have used very simple example and simple English words that you can pick quickly concept about functions.
#inshallah.
X std maths - Relations and functions (ex 1.1), Maths, IX std Maths, Samacheerkalvi maths, II year B.Ed., Pedagogy, Mathematics, Relation, Functions, Cartesian product, ordered pair, definition of cartesian product, standard infinite set, cartesian product of three sets,
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Cartesian product of two sets
1. CARTESIAN PRODUCT
Janak singh Saud
saudjanaksingh@gmail.com
https://images.app.goo.gl/CjPHqUg6ZxC3ehER6
6/19/2020 JANAK SINGH SAUD 1
2. 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
6/19/2020 JANAK SINGH SAUD 2
3. 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”.
6/19/2020 JANAK SINGH SAUD 3
4. 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
6/19/2020 JANAK SINGH SAUD 4
5. 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
BMW (BMW, White) (BMW, Red)
HONDA (HONDA, White)
NISSAN
B
A
× White Red
BMW (BMW, White) (BMW, Red)
HONDA (HONDA, White) (HONDA, Red)
NISSAN
B
A
× White Red
BMW (BMW, White) (BMW, Red)
HONDA (HONDA, White) (HONDA, Red)
NISSAN (NISSAN, White)
B
A
× White Red
BMW (BMW, White) (BMW, Red)
HONDA (HONDA, White) (HONDA, Red)
NISSAN (NISSAN, White) (NISSAN, Red)
6/19/2020 JANAK SINGH SAUD 5
7. 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
8. 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)}
6/19/2020 JANAK SINGH SAUD 8
9. 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
6/19/2020 JANAK SINGH SAUD 9
10. 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
6/19/2020 JANAK SINGH SAUD 10
11. 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}
6/19/2020 JANAK SINGH SAUD 11
12. 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}
6/19/2020 JANAK SINGH SAUD 12
13. 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
6/19/2020 JANAK SINGH SAUD 13
14. 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.
6/19/2020 JANAK SINGH SAUD 14
15. 5. SOLVE
• If A = {x : x ≤3 and x ∈ 𝑁}, then find A ×A
• If P = {x : 7≤x ≤10}, then find P ×P.
6/19/2020 JANAK SINGH SAUD 15
16. 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.
6/19/2020 JANAK SINGH SAUD 16
17. 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
6/19/2020 JANAK SINGH SAUD 17
18. 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:
6/19/2020 JANAK SINGH SAUD 18