Notes for math 141
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Exam2 review
Finite Mathematics
Linear programming Problems
Example 1.1. Minimize C = 2x + 5y, subject to
4x + y ≥ 40
2x + y ≥ 30
x + 3y ≥ 30
x≥0
y≥0
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Notes for math 141
2
Exam2 review
Finite Mathematics
Set
Example 2.1. Let U = {1, 2, 3, 4, a, b, c, d}, A = {1, 2, a, b}, B = {2, 3, a, b, c} and
C = {2, 4, a, c}, D={1,2}. Which elements are in A and which set is a (proper) subset
of A? Find Ac , A ∪ B, A ∩ C, A ∪ B ∪ C and A ∪ (B ∩ C)
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Exam2 review
Notes for math 141
Finite Mathematics
De Morgan Law
(A ∪ B)c = Ac ∩ B c
(A ∩ B)c = Ac ∪ B c
Distributive laws
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
The Number in the Union of Two Sets
For any finite sets A and B,
n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
If A and B are disjoint sets , then
n(A ∪ B) = n(A) + n(B)
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Notes for math 141
Exam2 review
Finite Mathematics
Example 2.2. A survey was conducted of College Station residents to determine
what activities they participated in during the 4th of July weekend. It was found
that 955 residents watched fireworks.
50 residents only went swimming.
528 residents participated in exactly two of these activities.
1168 residents watched fireworks or ate BBQ.
250 residents only watched fireworks and ate BBQ.
60 residents did not watch fireworks and did not eat BBQ.
425 residents watched fireworks, ate BBQ, and went swimming.
523 residents went swimming and ate BBQ.
How many residents did not eat BBQ?
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Notes for math 141
3
Exam2 review
Finite Mathematics
Multiplication principle
Example 3.1. Jack and Jill and 5 of their friends go to the movies. They all sit next
to each other in the same row. How many ways can this be done if
a) there are no restrictions?
b) Jill must sit in the middle?
c) Jill sits on one end of the row and Jack sits on the other end of the row?
d) Jack, Jill, or John sit in the middle seat?
e) Jack, Jill, and John sit in the middle seats?
f) Jack and Jill must sit next to each other?
g) Jill must not sit next to Jack?
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Notes for math 141
Exam2 review
Finite Mathematics
Example 3.2. Suppose we are playing the lottery in which we must choose 6 from
50 numbers.
a) How many different lottery picks could we choose if the order we choose our
numbers in does not matter?
b) How many ways are there to choose no winning numbers?
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Notes for math 141
4
Exam2 review
Finite Mathematics
Sample Space and Event
Example 4.1. Let the experiment be choosing a card out of a standard deck. Let E
be the event of drawing a club card; let F be the event of drawing a face card; let G
be the event of drawing a red card.
1. Determine the event E ∩ F . Are they mutually exclusive?
2. Determine the event E ∩ G. Are they mutually exclusive?
3. Determine the event (E ∪ F ∪ G)c .
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