key concepts
proposition: a statement that is true of false
propostional variable: a variable that represents a proposition, e.g., use p describe proposition Tom is a boy
negation of p: the proposition with truth value opposite to the truth value of p
logic operator: operator used to combine propositions
compound propostion: a proposition constructed by combining propositions using logical operators
truth table: a table displaying ALL the truth values of propositions
disjunction of p and q: p or q
conjunction of p and q： p and q
exclusive or of p and q: p xor q, exactly one of p and q is true
p implies q: p->q
contrapositive of p->q: negation(q) -> negation(p)
converse of p->q: q->p
inverse of p->q: negation(p)->negation(q)
p<->q biconditional: p->q and q->p
tautology: a compound proposition that is always true
contradiction: a compound proposition that is always false
contingency: a compound proposition that is sometime true and sometimes false
consistent compound propostions: compound propostions for which there is an assignment for truth values to the variables that makes all these propositions true
logically equivalent compound propositions: compound propositions always have the same truth values
predicate: part of a sentence that attributes a property to the subject
propositional function: a statement containing one of more variables that becomes a proposition when each of its variables is assigned a value or is bound by a quantifier
domain (or unniverse) of discourse: the values a variable in a propositional funciton may take
existential quantification of p(x): there is x such that p(x) is true
universal quantification of p(x): for every x, p(x) is true
logically equivalent expressions: expressions that have the same truth value no matter what proposition functions and domains are used
bound variable: a variable that is quantified
free variable: a variable that is not bound in a propositional function
scope of a quantifier: portion of a statement where the quantifier binds its variable
argument: a sequence of statements
argument form: a sequence of compound propositions involving propositional variables
premise: a satement in an argument, or argument form, other than then final conclusion
conclusion: the final statement in an argument or argument form
valid argument form: the truth of all the premises imply the truth of the conclusion
valid argument: an argument whose argument form is valid
rule of inference: a valid argument form that can be used in the demostration that arguments are valid
fallacy: an invalid argument form often used incorrectly as a rule of inference
circular reasoning or begging the question: reasoning where one or more steps are based on the truth of the statement being proved
theorem: a mathematical assertion that can be shown to be true
conjecture: a mathematical assertion proposed to be true, but that not been proved
proof: a demonstration that a theorem is true
axiom: a statement that is assumed to be true and that can be used as a basis for proving theorems
lemma: a theorem used to prove other theorems
vacuous proof: a proof that p->q is true is based on p is false
trival proof: a proof that p->q is trur is based on q is true
direct proof: a proof that p->q is true proceeds by showing that q must true when p is true
proof by contraposition: a proof that p->q is true proceeds by showing that negation(q)->negation(p)
proof by contradiction: a proof that p is true proceeds by showing that negation(p)->q, where is a contradiction
forward reasoning: direct proof, or proof by contraposition, or proof by contradiction
backward reasoning: when facing p->q, thinking r->q, where r is the last step leads to q
proof by case: (p1 or p2 or p3)->q is equivalent to (p1->q) and (p2->q) and (p3->q)
exhaustive proof: a proof that establishes a result by checking a list of all cases
without loss of generality: an assumption in a proof that makes it possible to prove a theorem by reducing the number of cases needed in the proof
counterexample: an element x such that p(x) is false
constructive existence proof: a proof that an element with a specified property exists that explicitly finds such an element
noconstructive existence proof: a proof that an element with a specified property exists that doesn't explicitly find such an element, usually by proof by contradiction, or claim exists in a small set
uniqueness proof: a proof that there is exactly one element satisfying a specified propery.
Results
logical equivalence (laws about and, or, negation)
De Morgan's law for quantifiers (negation of quantifiers)
rules of inference for proposition (8 laws)
rules of inference for quantified statement (unversal instantiation, unversal generalization, existential instantiation, existential generalization)
others
A collection of logical operators is called functionally complete if every compound proposition is logically equivalent to a compound proposition involving only these logical operators.
and, or, negation forms a functionally complete collection of logic operator, since a compound proposition with its truth table can be formed by taking the disjunction of conjunctions of variables or their negations, with one conjunction included for each combination of values for which the compound proposition is true.