Understanding Discrete Structures: Propositional Equivalences, Predicates, and Quantifiers, Assignments of Discrete Structures and Graph Theory

Download Understanding Discrete Structures: Propositional Equivalences, Predicates, and Quantifiers and more Assignments Discrete Structures and Graph Theory in PDF only on Docsity!

CECS 228

Discrete Structures with

Computer Science

Instructor: Mr. Jeho Park

1.2 Propositional

Equivalences

Always true no matter what the truth values of the propositions that occur in it. Always false Neither tautology nor contradiction

Tautology:

Contradiction:

Contingency:

Compound propositions that have the same truth

values in all possible cases are called logically

equivalent -- denoted by.

A lot of Laws to remember -- p. 24

An important type of step used in a mathematical argument is the replacement of a statement with another statement with the same truth value.

Quantifiers

P ( x ) has no truth value until the variable x is bound by

either

1) assigning it a value or

2) quantifying it.

Quantifiers Assignment of Values Let Q ( x ) denote “ x + y = 3” Q (2,1) Each of the following can be determined as T or F Q (3,2) Q (2,1) Q (3,2) ~[Q(2,1) Q(3,2)]

Quantifiers Universal Quantification of P ( x ): x P ( x ) x P ( x ) “for all x P ( x )” “for every x P ( x )” “There exists an x P ( x )” “There is at least one x P ( x )” Existential Quantification of P(x): Quantifiers Notations

Quantifiers Universal Quantification of P ( x ): x P ( x ) Quantifiers

Defined as:

P ( x 0 ) P ( x 1 ) P ( x 2 ) P ( x 3 )... for all x i in U

Example) Let P ( x ) denote x 2 ≥ x If U is real number, x, such that 0 < x < 1, False If U is real number, x, such that x > 1, True then x P ( x ) is then x P ( x ) is

Quantifiers Negating quantifiers ¬ x P ( x ) x ¬ P ( x ) ¬ x P ( x ) x ¬P ( x ) at least one false x P ( x ) all true x P ( x ) at least one true all false negation negation not all true none true Quantifiers

Quantifiers The Statement is true is false x P ( x ) x P ( x ) x ¬ P ( x ) x ¬ P ( x ) if for at least one x , P ( x ) is false if for all x , P ( x ) is false if for all x , P ( x ) is false if for all x , P ( x ) is true if for all x , P ( x ) is true if for at least one x , P ( x ) is true if for at least one x , P ( x ) is true if for at least one x , P ( x ) is false

Quantifiers Translate these statements into English, where R(x) is “x is a rabbit” and H(x) is “x hops” and universe of discourse consists of all animals. x(R(x) →H(x)) x(R(x) H(x)) x(R(x) →H(x)) x(R(x) H(x)) If an animal is rabbit, then that animal hops => Every rabbit hops Every animal is a rabbit and hops There exists an animal such that if it is a rabbit, then it hops There exists an animal that is a rabbit and hops => Some rabbits hop.

Group Exercise