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A proposition is a statement that can be true or false, forming the basis of propositional logic. Different types of propositions, logical operators, and truth tables are explored, along with key concepts like tautologies, contradictions, satisfiability, equivalence, and normal forms. Examples and explanations elucidate conjunctions, disjunctions, negations, implications, and other logical rules, essential for constructing proofs in logic. The content covers essential terminology, rules of inference, and methods for logical reasoning using propositions.
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What is a proposition? What is not? Proposition is a declarative statement that can be true or false Colgate University is liberal arts college. This is the year 2020. Albany is the capital of New York. 31 is a prime number. Every even integer greater than 2 can be written as the sum of two prime numbers. 2 + 2 = 4 Not a proposition How old are you? Stand up! I am too old to teach. x + 3 = 7
Combinations Albany is the capitol of New York and Dallas is the capital of Texas. If the temperature is 60 degrees, then George feels cold. That wall is red or it is blue. Today is not New Year’s eve.
Representing combinations We denote simple propositions by letters, usually p, q, r, etc. So we could write p = “Albany is the capital of New York.” and q = “Dallas is the capital of Texas.” We represent “and” by “^” and “or” by “v”. We represent “not” by “~”. With the above then p ^ q means “Albany is the capital of New York and Dallas is the capital of Texas” What does ~q mean? What does p v q mean?
Implication if p, then q If it is cold tonight, then it will snow. Let p = “it is cold tonight” and q = “it will snow” The above is then represented: if p, then q, or by p=>q Other examples??
Terminology • p ^ q is called conjunction, “p and q” • p v q is called disjunction, “p or q” • ~p is negation, “not p” • p => q is implication: “if p then q” • p <=> q means “p if and only if q" • We add p xor q Exclusive or (book uses + with a circle): “p or q but not both” • Each is fully defined by its truth table • Since each proposition, p, q, etc., is either true or false we can fully define the combinations is we specify whether they are true or false for every possible combination ofr p and q, etc.
Truth tables p ~p T F F T p q p ^ q T T T T F F F T F F F F p q p v q T T T T F T F T T F F F p q p => q T T T T F F F T T F F T p q p <=> q T T T T F F F T F F F T p q p xor q T T F T F T F T T F F F Note that for =>, when the antecedent (p) is false, the => is true, regardless of q. Why is this?
Why are we looking at these combinations of propositions?? • They are the basis of being able to construct proofs. • The study of logic goes back to the Greeks, so we will see some Greek names for rules of inference
Tautology is a statement that is always true • Useful for constructing proofs • Standard tautologies give us guides to reason logically • Can always check with truth tables
Examples: Law of excluded middle p v ~p (construct truth table) Modus ponens In English: if p implies q and p is true, then q is true. (p ^ (p => q) ) => q Modus tollens ( (p => q) ^ ~q ) => ~p
Example p q ((p=>q) ^ (~p=>q)) => q T T T T T T T F F F T T F T T T T T F F T F F T
Try these p q (p=>q)^(p=>r) <=> (p=>(q^r)) p q (p=>(q=>r)) <=> ((p^q)=>r)
Contradiction The opposite of a tautology is a contradiction, a proposition that is always false. Simplest example: p p^~p T F F F More interesting example: p q (p <=> q) ^ (p xor q) T T T F F T F F F T F T F F T F F T F F
Why are contradictions useful? Use for a proof method: If we can start with a hypothesis H and from that prove a contradiction C, then we have proved H=>C But since we know that C is always false, the contrapositive ~C=>~H shows us that ~H is always true.
Satisfiable In between the tautology and contradiction, a proposition is satisfiable if there is some assignment of truth values for its constituents that makes it true. Example: p q p^~q T T F T F T satifiable F T F F F F
Equivalence Two propositions P and Q are equivalent if P <=> Q is a tautology. In other words P and Q are equivalent if they have the same truth table. We write P=Q. Equivalence can be used in proofs. If we have established the truth of P and we know P and Q are equivalent, then we have established the truth of Q.
Contrapositive p=>q is equivalent to ~q=>~p Do not confuse with The converse of p=>q which is q=>p, not equivalent or the inverse of p=>q, which is ~p=>~q, also not equivalent
Example proof using equivalence To prove: (p^q)=>(pvq) is a tautology (p^q)=>(pvq) = ~(p^q) v (pvq) = (~pv~q)v(pvq) = (~pv~q) v(qvp) commutative = (~p v (~q v q) v p) associative = (~p v p v (~q v q) commutative = (~p vp) v(~qvq) = T v T = T
Disjunctive normal form Given any truth table, there is a proposition in this form that matches it A disjunction ( v ) of several conjunctions ( ^ ).
Truth Table to Normal Form A B C expression 1 1 1 1 1 1 0 1 1 0 1 1 1 0 0 0 0 1 1 1 0 1 0 0 0 0 1 0 0 0 0 0 A ^ A ^ B ^ C B ^ ~C A ^ ~B ^ C ~A ^ B ^ C (A^B^C) v (A^B^~C) v (A^~B^C) v (~A^B^C)
Conjunctive normal form Represent an expression (truth table) as the Conjunction ( ^ ) of several disjunctions ( v )
