## Discrete Test 1

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AnAngrySpring  on January 26, 2012

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# Discrete Test 1

 SubsetA⊆B, "A is a subset of B" means ∀x(x∈A=>x∈B) a set whose members are members of another set If A is not a subset of B: A⊄B
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#### Definitions

Subset A⊆B, "A is a subset of B"
means ∀x(x∈A=>x∈B)
a set whose members are members of another set

If A is not a subset of B: A⊄B
Proper subset If A⊆B, but A≠B (meaning there is something in B which is not in A)
A subset that does not contain every element in another set.
Relative complement The complement of A relative to B, written B - A is the set of all elements in B that are not in A.
Cartesian Product the set of elements common to two or more sets
Propositional function a statement containing one or more variables that becomes a propositions when each of its variables is assigned a value or is bound by a quantifier
Universal quantification "P(x) is true for all values x in the universe of discourse."

AxP(x)
Existential quanti cation "There exists an element x in the universe of discourse such that P(x) is true."

ExP(x)
proposition is a declarative sentence that is either
true (denoted either T or 1) or
false (denoted either F or 0).
conjunction "p and q", p ^ q
disjunction "p or q", p V q
exclusive or "exactly one of p or q", \p xor q", p (+) q
implication "if p then q", p --> q
biconditional "P if and only if q", p <--> q.
P ^ Q only T if both P and Q are T
P V Q only F is both P and Q are F
p (+) q Only F when p and q are the same answer
p --> q Only false if p is T and q is F
converse switch p and q
inverse ~p --> ~q
contrapositive ~q --> ~p
p <--> q Only true when both p and q are the same
p|q Only F if p and q are T
p(down arrow)q Only T if p and q are F
Image x
pre-image y
xA(s) = 1 if s is a member of A
tautology proposition that is always true

pV~p

p ^ ~p
contingency neither a tautology nor a contradiction
logically equivalent if r <-->s is a tautology
Identity Laws(p ^ T = p)
(p ^ F = F)
(p V T = T)
(p V F = p)
Domination Laws P ∨ T ≡ T
P ∧ F ≡ F
Idempotent Laws P ∨ P ≡ P
P ∧ P ≡ P
Double Negation Law ¬(¬P) ≡ P
Commutative Laws P ∨ Q ⇔ Q ∨ P
P ∧ Q ⇔ Q ∧ P
Associative Laws (P ∨ Q) ∨ R ≡ P ∨ (Q ∨ R)
(P ∧ Q) ∧ R ≡ P ∧ (Q ∧ R)
Distributive Laws p∨(q∧r) ≡ (p∨q)∧(p∨r)
p∧(q∨r) ≡ (p∧q)∨(p∧r))
De Morgan's Laws ~ (P and Q) = ~ P or ~ Q
~ (P or Q) = ~ P and ~ Q
Absorption Laws P ∨ ( P ∧ Q ) ⇔ P
P ∧ ( P ∨ Q ) ⇔ P
Implication Law (P --> Q) ⇔ (~P ∨ Q)
unique existential quanti cation "There exists a unique element x in the universe of discourse such that P(x) is true."

E!xP(x)

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