MiniCourse CMPE16: Discrete Mathematics

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Mini-CourseCMPE16Discrete Mathematics

• Presented by Christopher Boswell

Lecture Material by Prof. Patrick Tantalo, Slides
by Janelle Yong
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Discrete Mathematics

• A systematic approach of solving equations that
  can be applied to computer science and
  engineering
• Helps to mathematically understand the way a
  computer thinks

Lecture Material by Prof. Patrick Tantalo, Slides
by Janelle Yong
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Logic

• A proposition is a statement which is (or can be
  evaluated as) true or false
• Examples
• It is raining today.
• 2 3 5
• 2 3 7
• true T false F
• Proposition Variable can use any variable to
  replace the proposition statement (ex. p, q, or
  r)

Lecture Material by Prof. Patrick Tantalo, Slides
by Janelle Yong
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Logic

• Propositions can also be represented by a phrase.
  Examples
• p It is sunny today.
• q I will go to the park.
• r I will stay home.

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Logic

• Use logical operators
• Negation
• Conjunction
• Disjunction v
• Implication ?
• Biconditional ?

Lecture Material by Prof. Patrick Tantalo, Slides
by Janelle Yong
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Negation

• Not p
• Notation p
• Example
• If p It is sunny today.
• Then, p It is not sunny today.

Lecture Material by Prof. Patrick Tantalo, Slides
by Janelle Yong
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Truth Table

• Is used to tell whether a proposition is true or
  false based on all the possible input
  combinations
• For negation

Inputs Outputs
Lecture Material by Prof. Patrick Tantalo, Slides
by Janelle Yong
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Conjunction

• For propositions p and q, the output will be true
  if and only if (notation IFF) both p and q are
  true
• Notation p q

Lecture Material by Prof. Patrick Tantalo, Slides
by Janelle Yong
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Disjunction v

• For propositions p and q, the output will be true
  if either or both p or q are true
• Notation p v q

Lecture Material by Prof. Patrick Tantalo, Slides
by Janelle Yong
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Implication ?

• If-Then Statement
• p is the premise and q is the conclusion
• Notation p ? q
• Examples
• x 2 implies that x 1 3 OR If x 2,
  then x 1 3
• If I score 90 or higher in this class, then I
  will get an A.

http//www.math.niu.edu/richard/Math101/implies.p
df
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Implication ?

• In the case where p is true and q is false, the
  outcome is false
• Innocent until proven guilty stand-point
• A false statement implies ANYTHING

http//www.math.niu.edu/richard/Math101/implies.p
df
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Biconditional ?

• If and Only If IFF
• True if p and q have the same truth value
• Notation p ? q

Lecture Material by Prof. Patrick Tantalo, Slides
by Janelle Yong
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Compound Proposition

• A combination of propositional variables and
  logical operations
• Examples
• ( p ? q ) v ( q ? r )
• (p q)

Lecture Material by Prof. Patrick Tantalo, Slides
by Janelle Yong
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Logic

• Proposition sentences can then be turned into a
  combination of symbols
• Take
• p It is sunny today.
• q I will go to the park.
• r I will stay home.
• Example p ? q
• If it is sunny today, I will go to the park.

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Practice Problems

• p It is sunny today.
• q I will go to the park.
• r I will stay home.
• s I will go to the beach.

• Express the following propositions as a sentence
• 1. p ? (q s) 2. p ? r

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Practice Problems

• p It is sunny today.
• q I will go to the park.
• r I will stay home.
• s I will go to the beach.

Express the following sentences as a
proposition 3. If it is sunny today, I will not
stay home. 4. If it is sunny today, I will go the
park or the beach.
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Tautology

• A compound proposition which is true for every
  assignment to its variables
• In other words
• A proof whose outcome is always true

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Logical Equivalences

• Equal by definition
• An equal sign with three lines

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Logical Properties

• Identity
• Domination
• Idempotent
• Double Negation
• Commutative
• Associative
• Distributive
• DeMorgans

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Identity

• Assumptions that have been proven
• Things that you know are true
• Example
• Trig Identities?

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Domination

• If a statement is dominated by a true statement,
  it is logically equivalent to be true
• For a false statement, it is logically equivalent
  to be false

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Idempotent

• When applied to two equal values, it gives that
  value as the result
• Example
• The result of p or p is p.
• The result of p and p is also p.

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Double Negation

• Two wrongs make a right.

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Commutative

• The ability to change the order without changing
  the outcome
• Example
• 3 2 2 3

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Associative

• The order of the operations do not matter as long
  as the sequence of the operands are not changed
• Usually used with parenthesis
• Example
• ( 2 5 ) 27 2 ( 5 27 )

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Distributive

• Distribute the outside to the inside
• Example
• 2 ( 5x 7 ) 2 ( 5x ) 2 ( 7 ) 10x 14

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DeMorgans Laws
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Practice Problems
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Assignment

• 1. Which of these sentences are propositions?
  What are the truth values of those that are
  propositions?
• a) The California State bird is the road runner
• b) Jump up and down
• c) An ostrichs eyes are larger than its brain
• d) 235
• e) 223
• F) X79

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Assignment Continued

• 2. Let P, Q, and R be the propositions
• P You forget your watch
• Q You miss the bus
• R You get to class on time
• Express each of these propositions as an English
  sentence.
• a) P ? Q
• b) Q? R
• c) Q? R
• d) (P?Q) V (Q?R)
• e) P V Q V R

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Assignment Continued

• 3. Determine whether each of these conditional
  statements is true or false.
• a) If 112, then 356
• b) If 224, then 437
• c) If 214, then 339
• d) If monkeys can fly, then 11thrive

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Assignment Continued

• 4. Use truth tables to verify the associative
  law.
• (p V q) V r p V (q V r)
• 5. Show that this conditional statement is a
  tautology with and without truth tables
• p ? (p V q) ? q

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Questions?