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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:
¬!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
Translate each statement into English and determine its truth value.
(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.
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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- Summer '11