CSCE 222 CSCE222 Exam 1 – Texas A&M
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CSCE 222 CSCE222 Exam 1 – Texas A&M
Predicate
A function P from a set D to the set Prop of propositions is a _____. Domain The set D is the _____ of P Universal Quantifier ∀xP(x) – P(x) holds for all x in D Existential Quantifier ∃xP(x) – P(x) holds for some x in the domain D
Logical Equivalence
Two statements involving quantifiers and predicates are logically equivalent iff they have the same truth values no matter which predicates are substituted into the statements and no matter which domain is used.
Prove ∀x(P(x)⋀Q(x)) ≡∀xP(x) ⋀∀xQ(x)
Suppose ∀x(P(x)⋀Q(x)) is true.
Then for all a in the domain P(a)
Description
CSCE 222 CSCE222 Exam 1 – Texas A&M
Predicate
A function P from a set D to the set Prop of propositions is a _____. Domain The set D is the _____ of P Universal Quantifier ∀xP(x) – P(x) holds for all x in D Existential Quantifier ∃xP(x) – P(x) holds for some x in the domain D
Logical Equivalence
Two statements involving quantifiers and predicates are logically equivalent iff they have the same truth values no matter which predicates are substituted into the statements and no matter which domain is used.
Prove ∀x(P(x)⋀Q(x)) ≡∀xP(x) ⋀∀xQ(x)
Suppose ∀x(P(x)⋀Q(x)) is true.
Then for all a in the domain P(a) ⋀Q(a) is true. Hence, both P(a) is true and Q(a) is true. Since P(a) is true for all a in the domain, ∀xP(x) is true. Since Q(a) is true for all a in the domain, ∀xQ(x) is true. Hence ∀xP(x) ⋀∀xQ(x) is true.
De Morgan’s Laws ¬∀xP(x) ≡∃x¬P(x) and ¬∃xP(x) ≡∀x¬P(x) Prove ¬∀xP(x) →Q(x)) ≡∃x(P(x) ∧¬Q(x)) ¬∀xP(x) →Q(x)) ≡∃x¬(P(x) →Q(x)) ≡ ∃x¬(¬P(x) ∨Q(x)) ≡ ∃x(¬¬P(x) ∧¬Q(x)) ≡ ∃x(P(x) ∧¬Q(x))
Valid Arguments
CSCE 222 CSCE222 Exam 1 – Texas A&M
CSCE 222 Discrete Structures for Computing School: Texas A&M University (Texas A&M) *Professor:math, Lee, KLAPPENECKER, Klappenacker, n… Documents (404) Q&A (32) Textbook Exercises (20+)
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CSCE 222 – Spring 2014 Register Now hw7.pdf Textbook Exercises Discrete Mathematics and Its Applications image Discrete Mathematics and Its Applications Ch 1, Section 1.1 Propositional Logic, Exercise 1The given sentence is a preposition as it is declarative and states a fact.
CSCE 222 CSCE222 Exam 1 – Texas A&M
Truth value of the…Discrete Mathematics and Its Applications Ch 2, Section 2.1 Sets, Exercise 1The given set contains the values of x which are the solutions of the equation {{x}^{2}}=1 .Discrete Mathematics and Its
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CSCE 222 CSCE222 Exam 1 – Texas A&M
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