FinalReview

# FinalReview

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Unformatted text preview: the preceding statements (called premises). In other words, in a valid argument it is impossible that all premises are true but the conclusion is false. 8 Modus Ponens The tautology (p ⋀ (p->q)) -> q is the basis for the rule of inference called “modus ponens”. p p -> q ------∴q 9 Modus Tollens ¬q p -> q -----∴¬ p “The University will not close on Wednesday.” “If it snows on Wednesday, then the University will close.” Therefore, “It will not snow on Wednesday” 10 Formal Argument Argument ¬p ∧ q r→p ¬r → s s→t ∴t 1) 2) 3) 4) 5) 6) 7) 8) ¬p ∧ q ¬p r→p ¬r ¬r → s s s→t t Hypothesis Simpliﬁcation of 1) Hypothesis Modus tollens using 2) and 3) Hyposthesis Modus ponens using 4), 5) Hypothesis Modus ponens using 6), 7) 11 Proofs Direct Proof Proof by Contradiction Proof by Induction - weak induction - strong induction - structural induction - transfinite induction (= induction over well-ordered sets) 12 Set Theory You should be familiar with - the builder notation - subsets, equality of sets - union, intersection, set difference, complement - cartesian product, power set - cardinality of sets (when is |A| <= |B|?) - countable and uncountable sets 13 De Morgan Laws A∩B =A∪B P roof : A ∩ B = {x | x ￿∈ A ∩ B } by deﬁnition of complement = {x | ¬(x ∈ A ∩ B )} = {x | ¬(x ∈ A ∧ x ∈ B )} by deﬁnition of intersection = {x | ¬(x ∈ A) ∨ ¬(x ∈ B )} de Morgan’s law from logic = {x | (x ￿∈ A) ∨ (x ￿∈ B )} by deﬁnition of ￿∈ = {x | x ∈ A ∨ x ∈ B } by deﬁnition of complement = {x | x ∈ A ∪ B } by deﬁnition of union =A∪B 14 Functions Let f: A -> B be a function. We call - A the domain of f and - B the codomain of f. The range of f is the set f(A) = { f(a) | a in A } 15 Functions Let A and B be sets. Consider a function f: A-> B. - When is f surjective? - When is f injective? - When is f bijective? 16 Floor and Ceiling Functions The floor function : R -> Z ass...
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## This test prep was uploaded on 03/24/2014 for the course CSCE 222 taught by Professor Math during the Fall '11 term at Texas A&M.

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