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Predicates and Quantifiers

CSE 2813 Discrete Structures. Predicates. A predicate is a statement that contains variables.Example: P(x) : x > 3 Q(x,y) : x = y 3 R(x,y,z) : x y = z. CSE 2813 Discrete Structures. Predicates. A predicate becomes a proposition if the variable(s) contained is(are)Assigned specific

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Predicates and Quantifiers

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    1. CSE 2813 Discrete Structures Predicates and Quantifiers Section 1.3

    2. CSE 2813 Discrete Structures Predicates A predicate is a statement that contains variables. Example: P(x) : x > 3 Q(x,y) : x = y + 3 R(x,y,z) : x + y = z

    3. CSE 2813 Discrete Structures Predicates A predicate becomes a proposition if the variable(s) contained is(are) Assigned specific value(s) Quantified P(x) : x > 3. What are the truth values of P(4) and P(2)? Q(x,y) : x = y + 3. What are the truth values of Q(1,2) and Q(3,0)?

    4. CSE 2813 Discrete Structures Quantifiers Two types of quantifiers Universal Existential Universe of discourse - the particular domain of the variable in a propositional function

    5. CSE 2813 Discrete Structures Universal Quantification P(x) is true for all values of x in the universe of discourse. ?x P(x) “for all x, P(x)” “for every x, P(x)” The variable x is bound by the universal quantifier, producing a proposition

    6. CSE 2813 Discrete Structures Example U = {all real numbers}, P(x): x+1 > x What is the truth value of ?x P(x) U = {all real numbers}, Q(x): x < 2 What is the truth value of ?x Q(x) U = {all students in CSE 2813} R(x) : x has an account on banner What does ?x R(x) mean?

    7. CSE 2813 Discrete Structures For universal quantification P(x) ? P(x1) ? P(x2) ? … ? P(xn) If the elements in the universe of discourse can be listed, U = {x1, x2, …, xn} ?x P(x) ? P(x1) ? P(x2) ? … ? P(xn) Example U = {positive integers not exceeding 3} and P(x): x2 < 10 What is the truth value of ?x P(x) P(1) ^ P(2) ^ P (3) T ^ T ^ T T

    8. CSE 2813 Discrete Structures Existential Quantification P(x) is true for some x in the universe of discourse ?x P(x) “for some x, P(x)” “There exists an x such that P(x)” “There is at least one x such that P(x)” The variable x is bound by the existential quantifier, producing a proposition

    9. CSE 2813 Discrete Structures

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