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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 wellordered 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.
 Fall '11
 math

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