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Unformatted text preview: McCombs Math 381 Predicates and Quantifiers Important Vocabulary and Notation: Predicate: The part of a propositional statement that refers to a property that the statement of the subject can have. e.g. P : “is a perfect square”. Propositional Function: e.g. Quantifier: P ( x ) = the statement comprised of the predicate P and the subject x . P ( x ) : “ x is a perfect square.” A description that expresses the extent to which a predicate is true over a range of elements. 1. Existential Quantifier: !x “There exists an element for which it is true that...” 2. Universal Quantifier: !x “For every element it is true that...” Nested Quantifiers: We say that two quantifiers are nested if one is within the scope of the other. Important Note: Orders Matters! e.g. !x"y ( x # 2 y = 0 ) is not the same as !y"x ( x # 2 y = 0 ) Negating Quantifiers: 1. ¬!xP ( x ) = "¬P ( x ) 2. ¬!xP ( x ) = "¬P ( x ) Examples: 1. P(m,n): “m ≤ n”, where the universe of discourse for m and n is the set of nonnegative integers. Translate each statement into English and determine its truth value. (i) ∀nP(0,n) True (ii) ∃n∀mP(m,n) False (iii) ∀m∃nP(m,n) True McCombs Math 381 Predicates and Quantifiers 2. Negate the given statement. Express your answer so that no negation precedes a quantifier or a compound statement. (i) !w"a!f ( P ( w, f ) # Q ( f , a )) !w"a!f ( ¬P ( w, f ) # ¬Q ( f , a )) (ii) 3. ( !" > 0 #\$ > 0 !x ( 0 < x % a < \$ & !" > 0 #\$ > 0 !x 0 < x % a < \$ & f ( x ) % L < e f ( x) % L ' e ) ) The variables x and y represent real numbers, and E(x) : x is even G(x) : x > 0 I(x) : x is an integer. Write the statement using these predicates and any needed quantifiers. (i) Some real numbers are not positive. ∃x¬G(x). (ii) No even integers are odd. 4. ¬∃x(I(x) ∧ E(x) ∧ ¬E(x)]). Consider the predicates F(A): “A is a finite set” S(A,B): “A is contained in B” where the universe of discourse consists of all sets. Translate each statement into symbols. (i) Not all sets are finite. ∃A¬F(A). (ii) Every subset of a finite set is finite. ∀A∀B[(F(B) ∧ S(A,B)) → F(A)]. (iii) No infinite set is contained in a finite set. ¬∃A∃B(¬F(A) ∧ F(B) ∧ S(A,B)). (iv) The empty set is a subset of every finite set. ∀A[F(A) → S(∅,A)]. McCombs Math 381 Predicates and Quantifiers 5. The variable x represents students, y represents courses, and T(x,y) means “x is taking y”. Match the English statement with all its equivalent symbolic statements in the given list. 1. ∃x∀yT(x,y) 2. ∃y∀xT(x,y) 3. ∀x∃yT(x,y) 4. ¬∃x∃yT(x,y) 5. ∃x∀y¬T(x,y) 6. ∀y∃xT(x,y) 7. ∃y∀x¬T(x,y) 8. ¬∀x∃yT(x,y) 9. ¬∃y∀xT(x,y) 10. ¬∀x∃y¬T(x,y) 11. ¬∀x¬∀y¬T(x,y) 12. ∀x∃y¬T(x,y) A. Every course is being taken by at least one student. 6 B. Some student is taking every course. 1,10,6 C. No student is taking all courses. 12 D. There is a course that all students are taking. 2,3 E. Every student is taking at least one course. 3 F. There is a course that no students are taking. 7 G. Some students are taking no courses. 5, 8, 11 H. No course is being taken by all students. 9 I. Some courses are being taken by no students. 7 J. No student is taking any course. 4,7 ...
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