Download Logic Terms and Concepts and more Quizzes Discrete Mathematics in PDF only on Docsity!
Statement
A statement is a sentence that is either true or false. Opinions and ambiguous sentences (mathematical or otherwise) are not statements. x+y = 13 this can either be true or false and is not definitely one or the other so it is not a statement. TERM 2
DEFINITION 2 Means not. so ~p is the negation of p TERM 3
V
DEFINITION 3 Means or but this "or" implies it can be p or q or both in p V q TERM 4
xor
DEFINITION 4 Represented by a circle with a cross in it. This is also "or" but it means an exclusive or. in p xor q it means p or q but not both. TERM 5
^
DEFINITION 5 This means "and" it represents the situation where you have p and q
Logically equivalent
To be logically equivalent means to have identical truth values in a truth table TERM 7
Tautology
DEFINITION 7 A tautology means a statement that is always true. Represented by a bold t. TERM 8
Contradiction
DEFINITION 8 A statement that is always false; represented by a bold "c". TERM 9
If-Then
Statements
DEFINITION 9 A conditional statement with a hypothesis that leads to a conclusion. If the hypothesis is p and the conclusion q this statement is represented by p --> q TERM 10
p --> q equivalence
DEFINITION 10 p -- >q = ~p V q
Process of evaluation
Order to evaluate a statement in.[ ( ) ] ; [ ~ ] ; [ ^ V ] ; [ --> ] TERM 17
Biconditional
DEFINITION 17 In logic and mathematics, the logical biconditional is the logical connective of two statements asserting "p if and only if q", where q is a hypothesis (or antecedent) and p is a conclusion (or consequent).~q --> ~p where as p if q implies q --> p TERM 18
Sufficient Condition
DEFINITION 18 r --> swhere r is a sufficient condition for s TERM 19
Necessary Condition
DEFINITION 19 ~r --> ~sWhere r is a Necessary Condition for s. TERM 20
p q Biconditional
DEFINITION 20 p <==> q is equivalent to (p --> q) ^ (q --> p)If p and q are also equivalent then the statement p<-->q is a tautology
Argument
a sequence of statements called premises which lead to a final statement, the conclusion. For an Argument to be valid, when all of the premises are true the conclusion is always true. This is best shown in a truth table. TERM 22
Premise
DEFINITION 22 A premise is a condition given in an argument which leads to a conclusion. In truth tables for arguments the conclusion is only considered if all premises are true! TERM 23
Modus Ponens
DEFINITION 23 Major Premise : p --> qMinor Premise: p is trueTherefore q is true TERM 24
Modus Tollens
DEFINITION 24 Major Premise: p --> qMinor Premise: ~ qConclusion : ~p TERM 25
Generalization
DEFINITION 25 Premise: PConclusion: P V Q
Domain
The set of all values that can be substituted into a predicate TERM 32
Truth Set
DEFINITION 32 A set of all values in a domain where the predicate is true.If given a polynomial with variables solve the polynomial using quadratic equation to show how the truth set was found. TERM 33
Quantifiers
DEFINITION 33 Universal and Existential A word that shows the quantity of possible values for a variable in a statement. TERM 34
Universal Quantifier
DEFINITION 34 Key Words:for all, every, each, any, arbitrary, each, given any TERM 35
Z
DEFINITION 35 This is for Integers a + or - can define it for positive or negative integers only
R
Real numbersThis includes all Rational and irrational numbers(such as pi) TERM 37
C
DEFINITION 37 Complex numbers, this includes all integersnumbers represented as a+bi TERM 38
Q
DEFINITION 38 Rational numbersNumbers that can be written as a fraction of two integers TERM 39
Existential Quantifiers
DEFINITION 39 there exists, there is a, there is at least one, for at least one, we can find a, for some* most, this is a there exists some who don't TERM 40 DEFINITION 40 complex numbers contain all real numbers which contains all Rational numbers which has all integers which has all natural numbersnon real numbers such as i are in complex numbers too
