What is Discrete Mathematics

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What is Discrete Mathematics?
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From http//dictionary.oed.com Discrete adj.
1. a. Separate, detached from others,
individually distinct. Opposed to continuous. 2.
a. Consisting of distinct or individual parts
discontinuous.
From http//mathworld.wolfram.com/ the branch of
mathematics dealing with objects that can assume
only distinct, separated values. in contrast
with "continuous mathematics," which is the
branch of mathematics dealing with objects that
can vary smoothly.
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Logic and proof

• What information do we need to know to determine
  if a statement is true or false?
• What constitutes a valid proof?
• How can we determine if two statements are
  equivalent, i.e., have the same meaning?

If m2 is even, then m is even. To prove this
we need to know (i) what does it mean
(definition) to be even or odd, (ii) that if an
integer n is not even, then it is odd, and (iii)
that this statement is equivalent to the
statement if m is odd, then m2 is odd.
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Sets

• What is a set and how can we represent a set?
• Are there restrictions on what can constitute a
  set?
• How can we combine sets?
• What kind of algebraic properties do set
  operations satisfy?

One floor of a dormitory houses 42 students.
Four of these students are taking UC 110, UC 140
and UC 160, 8 are taking 110 and 140, 5 are
taking 110 and 160, 7 are taking 140 and 160, 17
are taking 110, 18 are taking 140 and 14 are
taking 160. How many are taking none of these
three courses?
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Graph Theory

• How can we represent relationships between sets
  of discrete objects?
• What is the most efficient ways of performing
  some task (for example, plowing snowy streets
  with the fewest number of passes per street)?

A train carries highly volatile chemicals. In
the event of a derailment, we must take care that
certain chemicals are not placed in adjacent
tankers. Can we arrange the cars in a safe way
and if so, how?
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Enumeration

• How many ways are there to perform some given
  task?
• How can we generate each way of performing a
  task and what is the most efficient manner for
  doing so?
• What is the probability that some event will
  occur?

In an Ohio Super Lotto drawing, six different
numbers are selected from the numbers 1-49, plus
a bonus ball is drawn. To win the entire
jackpot, you must select the six correct numbers.
Prizes are also awarded if you select 3, 4 or 5
of the six numbers and the bonus ball or if you
select 4 or 5 numbers without the bonus ball.
What are your chances of winning any given prize?
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Other Topics

• Matrices
• Functions
• Relations
• Number Theory
• Algorithms

Class composition

• Computer Science 8 students
• Integrated Mathematics 3 students
• Mathematics 7 students
• Middle Childhood Education (Mathematics) 7
  students
• Other 1 student