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Outline. 1. Predicates. 2. Quantifiers. 3. Equivalences. 4. Nested Quantifiers. Richard Mayr (University of Edinburgh, UK). Discrete Mathematics. Chapter 1.4-1.5. 2 / 23
1.4 Predicates and Quantifiers. Predicate Logic. Predicate logic have the following features to express propositions: • Variables: x, y, z, etc. (the subject of a sentence), can be substituted with an element from a domain. • Predicates: P, M, etc. (the predicate of a sentence). • Domain: the collection of values that a variable can
Predicates. Quantifiers. Scope, bound variables. Nested quantifiers. Distributivity. Negation. Examples p. 2. Predicates. Propositional logic, studied previously, cannot adequately express the meaning of all statements in mathematics and in natural lan- guage. Examples? “x is greater than 5.” “x is greater than y.” “n is a prime
Interchange the values of two variables x and y. 1. Temp := x. 2. x := y. 3. y := temp. ? How to verify the program? ? Assume precondition P(x,y) holds. P(x,y): x=a, y=b. True. ? Check the values of variables after each step of the program. 1. Temp := x x=a, y=b, temp=a. 2. x := y x=b, y=b, temp=a. 3. y := temp x=b, y=a, temp=a.
1.2. PREDICATES, QUANTIFIERS. 11. 1.2. Predicates, Quantifiers. 1.2.1. Predicates. A predicate or propositional function is a state- ment containing variables. For instance “x + 2 = 7”, “X is American”,. “x<y”, “p is a prime number” are predicates. The truth value of the predicate depends on the value assigned to its variables.
PREDICATES AND QUANTIFIERS. 45. 3. Predicates and Quantifiers. 3.1. Predicates and Quantifiers. Definition 3.1.1. A predicate or propositional function is a description of the property (or properties) a variable or subject may have. A proposition may be created from a propositional function by either assigning a value to
existencial quantification: ?xP(x) says. “there is one or more element under consideration for which the predicate P is true.” Under the domain of natural numbers, ?xP(x) is true, since for instance P(7) is true. Predicate calculus: area of logic dealing with predicates and quantifiers. CSI2101 Discrete Structures Winter 2010:
Predicates and Quantifiers. 1. 22c:19, Chapter 2. Hantao Zhang. Terminology review. • Proposition: a statement that is either true or false. – Must always be one or the other! Example: “The sky is red”. 2. – Example: The sky is red. – Not a proposition: x + 3 > 4. • Boolean variable: A variable (usually p, q, r, etc.) that represents
1.3 Predicates and Quantifiers 21. Predicates and Quantifiers. HNT? J? THC. N. Statements involving variables, such as. “x > 3,” “x = y + 3.” and “x + y = z.” are often found in mathematical assertions and in computer programs. These statements are neither true nor false when the values of the variables are not specified.
Predicate. Logic and. Quantifiers. CSE235. Predicate Logic and Quantifiers. Slides by Christopher M. Bourke. Instructor: Berthe Y. Choueiry. Spring 2006. Computer Science & Engineering 235. Introduction to Discrete Mathematics. Sections 1.3–1.4 of Rosen firstname.lastname@example.org. 1/1. Notes. Predicate. Logic and.
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