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Applications of Venn Diagrams
Union rule: n(A
B) = n(A) + n(B) - n(A
Given n(A) = 6 n(A
B) = 2 n(A
B)
B) = 25
Determine n
(B)
_
Given
n(A
30
B) =
n(A) =
12
n(B) =
5
not
possible
_
Given
n(A
30
B) =
n(A) =
25
n(B) =
17
_
Given
n(A) = 16
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Expected
Value
Recall the experiment of flipping a coin 4
times
HHHH
HHHT
HHTH
HTHH
THHH
HHTT
HTTH
TTHH
HTHT
THTH
THHT
HTTT
THTT
TTHT
TTTH
TTTT
Note:
Determine the probability of
getting
no
heads
1
head
2
heads
3
heads
4
heads
Create t
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Sets
A set is a collection of objects
Typical notation: A = cfw_1,5,8,9
Order is not important: A = cfw_1,8,5,9
Elements of a set: 5
Subsets: cfw_1,5
A7
A
A
Empty set: set with no elements cfw_ or
The empty set is a subset of every
se
M1243_2
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Applications of Linear Programming
A company makes color television sets. It produces bargain sets for
$100 profit and a deluxe set for $150 profit. On the asembly line the
bargain set requires 3 hours and the deluxe set takes 5 hours.
Systems of Linear Equations
With a system of two linear equations, there are three possibilities:
The lines might
intersect at exactly one point
be parallel and distinct
be parallel and coincide
Solve the system:
2x + 3y = 8
3x - 2y = -1
or
The system ca
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Solving Systems of linear Equations II
Solve the
system:
x + 2y - 3z = - 2
3x - y - 2z = 1
2x + 3y - 5z = - 3
The first row
means:
The second row
means:
Solutions of the
form:
Mathcad
check:
Solve the
system:
x+y+z=1
3x - y - z = 4
x +
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Addition and Subtraction of Matrices
Size: row x column
Square matrix: rows = columns
Individual
elements:
Each
column:
Switches the rows and
columns
Transpose:
Each
row:
Addition:
Subtraction:
Note that in order to add or subtract mat
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Linear Programming
A Graphical Approach
The maximum or minimum value of the objective
function
will be at a corner point of the feasible region.
Maximize: z = 2x + 3y
subject
to
sketch the feasible
region
determine the corner
points
y-
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Graphing Inequalities
Sketch the graph of 2x + y <
4
Sketch a graph
of:
If there is more than one inequality, the region where they overlap is called the feasible
region.
Sketch the feasible
region:
3x + y < 20
x + y < 12
x>0
y>0
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