MAT 251 Discrete Mathematics

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MAT 251 Discrete Mathematics

• Logic and Proofs

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Section 1.3 Predicates and Quantifiers

• Def Statements involving variables such as
• x y 5, y 2x 7, x gt 4
• are statements with two parts. They each have
  a subject of the statement and then they have the
  predicates, which refers to the property of the
  subject can have.

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Section 1.3 Predicates and Quantifiers

• These statements are not propositions until we
  assign values to the subjects. We introduce the
  propositional funtion notation as
• let P(x) x lt 7.
• Then we can ask is P(10) true?
• Let Q(x,y) y 2x 7
• Then we can ask is P(0, 1) true?

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Section 1.3 Predicates and Quantifiers

• Def Quanitification is another way to create
  propositions from propositinal functions.
  Quanitification expresses the extent to which a
  predicate is true over a range of elements.
• The words we use are all, some, many, none
  and few.
• Example Every function that is differentiable
  at a is also continuous at a.

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Section 1.3 Predicates and Quantifiers

• Def The universal quantification of P(x)
• is a statement
• P(x) is true for all values of x in the domain.
• Notation for
  all/every x P(x)
• An element for which P(x) is false is called a
• counterexample of

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Section 1.3 Predicates and Quantifiers

• A Universal Quantification is true if P(x)
• is true for all x in the domain, and
• It is a false statement if P(x) is not
• always true when x is in the domain.
• Example Let P(x) be x gt 0. Determine if
  
• is true,
• where the domain is all real numbers.

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Section 1.3 Predicates and Quantifiers

• Def The existential quantification of P(x) is
  the proposition
• There exists an element x in the domain such
  that P(x).
• Notation
• Domain must be clear! We read the above as there
  is an x st ..., there is at least one x st , for
  some x

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Section 1.3 Predicates and Quantifiers

• An existential quantification is true if you
• can find any instances for which P(x) is
• true. It is only false when there is no
• element x in the domain for which P(x) is
• true.
• Example Let Q(x) be x2 -1. What is the truth
  value of
• where the domain is all real numbers?

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Section 1.3 Predicates and Quantifiers

• There are other quantifications we can
• use. The one that is used most often is
• the Uniqueness Quantification.
• Notation
• There exists exactly one x st ,
• there is only one x st

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Section 1.3 Predicates and Quantifiers

• Note
• We may want to restrict domains.
• We may have variables that are bound and some
  that are free.
• The universal and existential quantifiers have
  higher precedence then the other logic operators.

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Section 1.3 Predicates and Quantifiers
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Section 1.3 Predicates and Quantifiers

• Example
• 1) All lions can roar.
• 2) Some chocolates are bitter.
• Express each of these statements using
• quantifiers. Then form the negation of
• the statement and then express the
• negation in simple English.

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Section 1.3 Predicates and Quantifiers

• The notes have been created with the use
• of Discrete mathematics and Its
• Applications, Sixth Edition by K. H. Rosen