Discrete Mathematics

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Discrete Mathematics

• Discrete means
• apart
• distinct
• away from each other
• not continuous

Examples
The (sound) pitch of a violin is continuous.
The (sound) pitch of a piano is discrete.
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An electric analogue clock shows continuous time.
A digital clock shows discrete time.
A conventional photograph shows almost continuous
shades of color.
A digital picture shows discrete shades of color
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Subsets of the Real numbers
A subset of real numbers is said to be discrete
if every member in the subset has a neighborhood
that separates it from all other members in the
subset.
For example, the above set (indicated by red
dots) is discrete (click to see the private
neighborhoods.
On the other hand, the set of fractions between 0
and 1 is not discrete.
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Subsets of the Real numbers
It is clear that every finite subset of the real
number is discrete. But there are infinite
discrete subsets as well, such as the set of
integers.
Therefore, finite mathematics is a part of
discrete mathematics. But discrete mathematics
contains more than just finite mathematics.
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Some characteristics of Discrete mathematics

• In discrete mathematics, we do not take limit as
  x approaches a certain number.
• We do not use approximation techniques to solve
  equations.

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• Why do we study discrete mathematics?
• there are lots of intrinsically discrete
  problems that cannot be solved by calculus
  type techniques.
• computers (in the present) can only handle
  discrete structures.
• a discrete model can be used to approximate a
  continuous model to a very high degree of
  accuracy. (such as digital photos and digital
  music)

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Chapter 1 The Logic of Compound Statements
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Mathematical Logic
In order to study discrete mathematics and
understand computer programming, one must have
some basic knowledge of mathematical logic.
Hence in any beginning course of discrete
mathematics, about 50 of the time is spent on
mathematical logic.
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• Mathematical Logic
• A set of precise rules that governs the
  operations of computers (and our mind).

Propositional Calculus
Predicate Calculus

• A proposition is a sentence that is either true
  or false but not both.(In particular, it cannot
  be a question.)
• Examples
• 2 2 5
• sin(p/6) 0.5

• A predicate is a sentence that contains
  variables, and when the variables are substituted
  by numbers or actual objects, it becomes a
  proposition.
• Examples
• x gt 4
• a2 b2 c2

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Predicate Calculus
Propositional Calculus
A proposition cannot have free variables. Proposit
ional calculus is analogous to Arithmetic where
we do not deal with variables
Predicate calculus on the other hand is analogous
to Algebra, which is more complex than arithmetic
but it requires the knowledge of arithmetic.
Note a proposition is also called a statement.
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More examples of Propositions
Determine whether each of the following is a
proposition
Yes.
1. Washington, D.C., is the capital of USA.
No.
2. Please read this carefully.
Yes.
3. All Martians like pepperoni on their pizza.
4. Jane forgot to bring her umbrella.
Yes.
5. Would you please pass me the salt?
No.