COT3100 Exam 1

Double Negation Law

!(!p)≡p

De Morgan's Law

!(p∧q) ≡ !p∨!q and !(p∨q) ≡ !p∧!q

Distributive Laws

(p ∨ (q ∧ r))≡ (p ∨ q) ∧ (p ∨ r), (p ∧ (q ∨ r))≡ (p ∧ q) ∨ (p ∧ r)

Associative Laws

(p∧q)∧r≡p∧(q∧r), (p∨q)∨r≡p∨(q∨r)

Logical Operator Precedence

1. Negation 2. Conjunction 3. Disjunction 4. Implication 5. Bicondictional

Steps for translating math into predicate logic

1. Rewrite statement to make the implied quantifiers and domains explicit 2. Introduce variables and specify the domain 3. write it out

Steps for Negating Nested Quantifiers

1. Use quantifiers to express English equivalent of statement 2. Use De Morgan's Laws to move negation as far inwards as possible

The truth value of ∀xP(x) and ∃xP(x) a dependent upon what?

Both the propositional function P(x) and on the domain U

When can looping through nested quantifiers not be carried out?

If the domains of the variables used are infinite

Question that must be asked to know if "You can switch order of quantifiers" or "You can distribute quantifiers over logical connectives"

Is this a valid equivalence?

A programming analogy for logical quantifiers?

Loops

What has higher precedence than all the logical operators?

Quantifiers [example: ∀xP(x)∨Q(x) means (∀xP(x)) ∨ Q(x) ] NOTE: Quantifiers quantify the thing they are closest to

Explain logical implication

The material conditional stating only that q is true when (but not necessarily only when) p is true, and makes no claim that p causes q (i.e. all is okay unless you break the contract)

When are two propositions equivalent?

Two propositions are equivalent if they always have the same truth value

define a lemma

a "helping theorem" or a result which is needed to prove a theorem

if the domain is finite (or infinite), a universally quantified proposition is equivalent to what?

a conjunction of propositions without quantifiers [U are integers 1,2,3 so ∀xP(x) ≡ P(1) ∧ P(2) ∧ P(3)]

Define a proposition

a declarative sentence that is either true or false

if the domain is finite (or infinite), an existentially quantified proposition is equivalent to what?

a disjunction of propositions without quantiers [U are integers 1,2,3 so ∃xP(x) ≡ P(1) ∨ P(2) ∨ P(3)]

define hypothetical syllogism

a logical argument that applies deductive reasoning to arrive at a conclusion based on two or more propositions (in this case conditional statements) that are asserted or assumed to be true ( (p→q) ∧ (q→r) → (p→r) )

Define contradiction

a proposition which is always false

Define tautology

a proposition which is always true

Define contingency

a proposition which is neither a tautology nor a contradiction

define a corollary

a result which follows directly from a theorem

define argument in propositional logic

a sequence of propositions (all but the final proposition are called premises with the last statement being the conclusion)

define resolution inference rule

a single valid inference rule that produces a new clause implied by two clauses containing complementary literals (A literal is a propositional variable or the negation of a propositional variable ((!p∨r) ∧ (p∨q)) →(q∨r) )

Define quantifier

a specifier for the domain of a predicate (most common are "for all" and "there exists" (i.e. all and some))

define theorem

a statement that an be shown to be true using definitions, other theorems, axioms (statements given as true), and rules of inference

define conjecture

a statement that is being proposed to be true (once true it becomes a theorem)

define disjunctive syllogism

a valid argument for which is a syllogism having a disjunctive statement as one of its premises ( !p ∧ (p∨q) → q )

define universal generalization (UG)

a valid inference rule stating that if P(x) has been derived, then ∀xP(x) can be derived

define universal instantiation (UI)

a valid rule of inference from a truth about each member of a class of individuals, to the truth about a particular individual of that class

define existential generalization (EG)

a valid rule of inference that allows one to move from a specific statement, or one instance, to a quantified generalized statement, or existential proposition

define existential instantiation (EI)

a valid rule of inference which says that, given a formula of the form (∃x)P(x) , one may infer P(c) for a new constant or variable symbol c. The rule has the restriction that the constant or variable c introduced by the rule must be a new term that has not occurred earlier in the proof.

What is an argument form

an argument that is valid no matter what propositions are substituted into its propositional variables

Quantifiers are said to _______ the variable x in these expressions

bind

How to get the converse of a proposition?

flip hypothesis and conclusion (q→p goes to p→q)

How to get the contrapositive of a proposition?

flip hypothesis and conclusion, then negate both (!q→!p goes to p→q)

When is a compound proposition unsatisfiable

if and only if its negation is a tautology

Statements involving predicates and quantifiers are logically equivalent.....

if and only if they have the same truth value for every predicate and every domain

Define consistent

if it is possible to assign truth values to the proposition variables so that each proposition is true

Common forms of writing implication

if p, then q \ p implies q \ if p, q \ p only if q \ q when/if p

define simplification (conjunction elimination) inference rule

if the conjunction A and B is true, then A is true, and B is true. The rule makes it possible to shorten longer proofs by deriving one of the conjuncts of a conjunction on a line by itself. ( (A∧B)→A )

When is an argument valid

if the premises imply the conclusion

Define satisfiable (for compound propositions)

is there is an assignment of truth values to its variables that make it true (not algebraic manipulation but substitution to make it work)

Define exclusive disjunction (or/ Xor)

logical operation that outputs true only when inputs differ (one is true, the other is false)

How to get the inverse of a proposition?

negate it (!p→!q goes to p→q)

Alternate forms of expressing the biconditional

p is necessary and sufficient for q / if p then q, and conversely / p iff q

Expression with variables are not....

propositions and therefore do not have truth values

Identity Laws

p∧T≡p, p∨F≡p

Negation Laws

p∨!p≡T, p∧!p≡F

Absorption Laws

p∨(p∧q)≡p, p∧(p∨q)≡p

Domination Laws

p∨T≡T, p∧F≡F

Independent Laws

p∨p≡p, p∧p≡p

Commutative Laws

p∨q≡q∨p, p∧q≡q∧p

define addition inference rule

the inference that if p is true, the p or q must be true ( p →(p∨q) )

define conjunction (adjunction) inference rule

the inference that if the proposition p is true, and proposition q is true, then the logical conjunction of the two propositions p and q is true. The rule makes it possible to introduce a conjunction into a logical proof.

Explain logical biconditional

the logical connective of two statements asserting "p if and only if q", where p is an antecedent and q is a consequent.

define modus ponens

the rule of logic stating that if a conditional statement ("if p then q ") is accepted, and the antecedent ( p ) holds, then the consequent ( q ) may be inferred ( (p∧(p→q))→q is the corresponding tautology)

define modus tollens

the rule of logic stating that if a conditional statement ("if p then q ") is accepted, and the consequent does not hold ( not-q ), then the negation of the antecedent ( not-p ) can be inferred ( (!q∧(p→q))→!p is the corresponding tautology)

Logical symbol for implication

→

Logical symbol for biconditional

↔

Symbol for "Universal Quantifier" (i.e. for all)

∀

Symbolically !∃xP(x) is equivalent to what?

∀x!P(x)

________ asserts P(x) is true for every x in the domain

∀xP(x)

Symbol for "Existential Quantifier" (i.e. There exists) [some]

∃

How to say P(x) is true for one and only one x in the domain? (both in logic and english)

∃!xP(x) / There is a unique x such that P(x)

Symbolically !∀xP(x) is equivalent to what?

∃x!P(x)

________ asserts P(x) is true for some x in the domain

∃xP(x)

Logical symbol for conjunction (and)

∧

Logical symbol for disjunction (or)

∨

Symbols to show compound propositions (proposition with more than one logical operation) are equivalent

≡ or <=>