discrete math midterms

Propositions

• Is a declarative sentence that is either true or false, but not both

Propositional logic

• Is the area of logic that deals with prepositions

Law of identity, Law of excluded middle, Law of non-contradiction

Three Aristotelian logic that is the bases of propositional logic

Law of Identity

• “A thing statement is itself“

Law of excluded middle

• “A statement is either true or false, but not both“

Law of non-contradiction

• “No statement is both true and false“

Compound propositions

• Are statements that are formed by combining one or more propositions

• Are formed from existing propositions using logical operators.

Compound statements

• These are groups of statements that are connected by words called “ connectives” such as “and“, “or“, “if-then“, “if only if“.

Negation Statement

• It is the denial or a given statement and uses the word no or not

Disjunction Statement

• In this statement, if one statement is true, then the compound statement is true, and if both statement are false then the compound statement is false.

Conjunction Statement

• In this statement, if one statement is false, then the compound statement is false, and if both statement is true, then the compound statement is true.

Conditional Statement

• In this statement, the first statemen is considered as the hypothesis and the second statement is considered as the conclusion.

• This statement is only false when the hypothesis is true but the conclusion is false.

Bi-conditional

• In this statement, the first statement is considered as the antecedent, while the second statement is considered as the consequent

• The statement is only true when both statements are true or both are false

Exclusive Or

• is a proposition that true when the when one of the statement is true and one of the statement is false, otherwise the proposition is false.

Tautology

• It is a compound proposition which is always true for every value of its propositional variables.

Contradiction

• It is a compound proposition which is always false for every value of is propositional variables

Contingency

• Is a compound proposition that is neither a tautology or a contradiction

Semantics

• It is what gives meaning or assigns truth values to the well-formed formula and allow us to understand and interpret them.

predicate

• Refers to a property that the subject of the statement can have

Quantifiers

• Expresses the extent to which a predicate is true over a range of elements

Predicate Calculus

• Area of logic that deals with predicates and quantifiers

Domain of discourse

• A particular domain where mathematical statements assert that a property is true for all values of variable

Universal Quantifier

• tells us that a predicate is true for every element under consideration

• Uses inverted A symbol

Existential Quantifier

• Tells us that there is one or more element for which the predicate is true.

• Uses reversed E as a symbol

Bound

• An occurrence of a variable when a quantifier is used on it

Free

• An occurrence of a variable that is not bound by a quantifier or set equal to a particular value

Set

• Is an unordered collection of objects called elements

Roster Method

• Is a way of describing a set where we list all the members or elements of the set

Set builder notation

• Way of describing a set where we characterize all the elements of the set by stating the property/properties that they must have to be members

N

• Natural number

• {0,1,2,3,4,….}

Z

• integers

• {…,-2,-1,0,1,2,3,…}

Q

• Rational Numbers

• {1/2,2.5,3.4,1/4}

R

• Real numbers

C

• complex numbers

Empty set

• It is a special set that has no element

• also known as “null set“

• Denoted by symbol “∅“ or {}

Subset

• Are set whose set are contain within another set

Power set

• It is the set of all the subset of the set

Union

• It is the set that contains the elements of that are either in set A or B

Intersect

• Set that contains the elements that are in both set A and set B

Disjoint

• Two sets that has an empty set as their intersection

Difference

• Is a set containing those elements that are in set A but not in set B

Universal set

• It is the complement of a set and contains the elements that is outside of a set.