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CSCI 115
Chapter 2
Logic
CSCI 115
§2.1
Propositions and Logical Operations
§2.1 – Propositions and Log Ops
• Logical Statement
• Logical Connectives
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Propositional variables
Conjunction (and: )
Disjunction (or: )
Negation (not: ~)
• Truth tables
§2.1 – Propositions and Log Ops
• Quantifiers
– Consider A = {x| P(x)}
– t  A if and only if P(t) is true
– P(x) – predicate or propositional function
• Programming
– if, while
– Guards
§2.1 – Propositions and Log Ops
• Universal Quantification – true for all
values of x
– x P(x)
• Existential Quantification – true for at least
one value
– x P(x)
• Negation of quantification
CSCI 115
§2.2
Conditional Statements
§2.2 – Conditional Statements
• Conditional statement: If p then q
–pq
– p – antecedent (hypothesis)
– q – consequent (conclusion)
• Truth tables
§2.2 – Conditional Statements
• Given a conditional statement p  q
– Converse
– Inverse
– Contrapositive
• Biconditional (if and only if)
– p  q is equivalent to ((p  q)  (q  p))
§2.2 – Conditional Statements
• Statements
– Tautology (always true)
– Absurdity (always false)
– Contingency (truth value depends on the values
of the propositional variables)
• Logical equivalence ()
CSCI 115
§2.3
Methods of Proof
§2.3 – Methods of Proof
• Prove a statement
– Choose a method
• Disprove a statement
– Find a counterexample
• Prove or disprove a statement
– Where do I start?
§2.3 – Methods of Proof
• Direct Proof
• Proof by contradiction
• Mathematical Induction (§2.4)
§2.3 – Methods of Proof
• Valid rules of inference
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((p  q)  (q  r))  (p  r)
((p  q)  p)  q
Modus Ponens
((p  q)  ~q)  ~p
Modus Tollens
~~p  p
Negation
p  ~~p
Negation
pp
Repitition
• Common mistakes – the following are NOT VALID
– ((p  q)  q)  p
– ((p  q)  ~p)  ~q
CSCI 115
§2.4
Mathematical Induction
§2.4 – Mathematical Induction
• If we want to show P(n) is true nZ,
n > n0 where n0 is a fixed integer, we can do this by:
i) Show P(n0) is true
• Basic step
ii) Show that for k > n0, if P(k) is true, then P(k + 1) is true
• Inductive step