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Math
Discrete Math
Language and Logic Definitions
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Terms in this set (36)
Proposition
any declarative sentence (ie it has both a subject and a verb) that is either true or false, but not both
Set
a well-defined collection of objects
Elements
objects or members of the set
R
the set of all real numbers
Z
the set of all integers
N
the set of all natural numbers
Q
the set of all rational numbers
E
the set of all even integers
O
the set of all odd integers
nZ
the set of all integers multiples of n (n E Z)
Predicate
any declarative sentence containing one or more variables, each variable representing a value in some prescribing set, called the universe, and which becomes a proposition when values from their respective universes are substituted for these variables
Proposition
Statement
Predicate
Open sentence
Ʌ
and (conjunction)
V
or (disjunction)
¬
negation
=>
implications
¬P
P is false
P => Q
P implies Q (if P is true, then Q is true"
in P => Q , P is called
assumption, or hypothesis, or antecedent
in P => Q , Q is called
conclusion, or consequent
implications
if, then
converse
the statement Q => P is called a converse of the statement P => Q
contrapositive
the statement (¬Q) => (¬P) is called a contrapositive of the statement P => Q
<=>
biconditional
P <=> Q
(P => Q) Ʌ (Q => P)
P <=> Q
"P is equivalent to Q" or "P if and only if Q"
Ξ
Logical Equivalence
Two compound statements are logically equivalent if
they have the same truth tables, which means they both are true or both are false
Double Negation Law
¬(¬P) Ξ P
Idempotent Laws
P V P Ξ P
P Ʌ P Ξ P
Commutative Laws
P V Q Ξ Q V P
P Ʌ Q Ξ Q Ʌ P
Associative Laws
P V (Q V R) Ξ (P V Q) V R
P Ʌ (Q Ʌ R) Ξ (P Ʌ Q) Ʌ R
Distributive Laws
P V (Q V R) Ξ (P V Q) V (P V R)
P Ʌ (Q V R) Ξ (P Ʌ Q) V (P Ʌ R)
De Morgan's Laws
¬(P V Q) Ξ (¬P) Ʌ (¬Q)
¬(P Ʌ Q) Ξ (¬P) V (¬Q)
Corollary
P <=> Q Ξ (¬Q) <=> (¬P)
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