discrete mathamatic by adeel
TRANSCRIPT
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Index
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Cartesian Products
Definition :
• Let A and B be two sets. The Cartesian
product of A and B, denoted AxB, is the
set of all ordered pairs (a,b) such that a
belongs A and b belongs B.
•
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{( , ) | and }.A B a b a A b B
Example: Finding Cartesian
Products
• Let A = {a, b}, B = {1, 2, 3}
• Find each set.
• a)AxB
• b)BxB
• Solution• a) {(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)}
• b) {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3),
• (3, 1), (3, 2), (3, 3)}
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Domain
• The domain of a function is the complete
set of possible values of the independent
variable.
• The domain is the set of all possible x-
values which will make the function "work“
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Range
• The range of a function is the complete
set of all possible resulting values of the
dependent variable
• The range is the resulting y-values we get
after substituting all the possible x-values.
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A relation is a set of ordered pairs.
{(2,3), (-1,5), (4,-2), (9,9), (0,6)}This is a
relation
The domain is the set of all x values in the relation
{(2,3), (-1,5), (4,-2), (9,9), (0,6)}
The range is the set of all y values in the relation
{(2,3), (-1,5), (4,-2), (9,9), (0,6)}
domain = {-1,0,2,4,9}
range = {-6,-2,3,5,9}
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Domain (set of all x’s) Range (set of all y’s)
1
2
3
4
5
2
10
8
6
4
A relation assigns the x’s with y’s
This relation can be written {(1,6), (2,2), (3,4), (4,8), (5,10)}4/5/2015 1 8
One- to - many
One- to - one many –to - one
Types of Relations
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A function f from set A to set B is a rule of correspondence that assigns to
each element x in the set A exactly one element y in the set B.
Set A is the domain
1
2
3
4
5
Set B is the range
2
10
8
6
4
A function f from set A to set B is a rule of correspondence that assigns to
each element x in the set A exactly one element y in the set B.
A function f from set A to set B is a rule of correspondence that assigns to
each element x in the set A exactly one element y in the set B.
This is a function ---it
meets our conditions
All x’s are
assigned
No x has more
than one y
assigned
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A good example that you can “relate” to is students in our maths class this
semester are set A. The grade they earn out of the class is set B. Each student
must be assigned a grade and can only be assigned ONE grade, but more than
one student can get the same grade (we hope so---we want lots of A’s). The
example show on the previous screen had each student getting the same grade.
That’s okay.
1
2
3
4
5
A
E
D
C
B
Is the relation shown above a function?NO
2 was assigned both B and E
A good example that you can “relate” to is students in our math class this semester
are set A. The grade they earn out of the class is set B. Each student must be
assigned a grade and can only be assigned ONE grade, but more than one
student can get the same grade (we hope so---we want lots of A’s). The example
shown on the previous screen had each student getting the same grade. That’s
okay.
How we identify function or Not
• If two or more pair have same value of X
then that condition is not function.
• (2,3),(2,4),(2,5),(2,1)
• If all the pair same value of y then no
effect on function.
• (2,3),(4,3),(6,3),(0,3)
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0,0
y
Y’
xX’
-1-2-3-4-5-6-1
-2
-3
-4
-5
654321
5
4
3
2
1
-6
6
2,4
2,3
2,1
2,5
(2,3),(2,4),(2,5),(2,1)
• (2,3),(4,3),(6,3),(0,3)
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0,0
y
Y’
xX’
-1-2-3-4-5-6-1
-2
-3
-4
-5
654321
5
4
3
2
1
-6
6
6,34,32,30,3
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0,0
y
Y’
xX’
-1-2-3-4-5-6-1
-2
-3
-4
-5
654321
5
4
3
2
1
-6
6
6,6
4,4
2,2
-2,-2
-4,-4
-6.-6
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0,0
y
Y’
xX’
-1-2-3-4-5-6-1
-2
-3
-4
-5
654321
5
4
3
2
1
-6
6
6,6
4,4
2,2
-2,6
-4,4
-6.2
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0,0
y
Y’
xX’
-1-2-3-4-5-6-1
-2
-3
-4
-5
654321
5
4
3
2
1
-6
6
0,4
4,0
-4,0
2,-6-2,-6
0,-4
Graph
• showing a relationship (usually between two set of numbers) by means of a line, curve, or other symbols.
• Typically, an independent variable is represented on the horizontal line (X-axis) and an dependent variable on the vertical line (Y-axis).
• The perpendicular axis intersect at a point called origin
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