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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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 Summer '11
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 Math

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