paper def Flashcards
(14 cards)
def: logically equivalent
two statements are logically equivalent when their truth values match up line for line on a truth table
def: FLT
where p is prime and a is an integer s.t. a isnt 0modp
a^(p-1) = 1modp
def: function
an assignment of a unique element of Y to each element of X
def: injective
f(A) = f(B) => a = b where a,b are elments of X
def: surjective
for all y ∈ Y there exists at least one x ∈ X such f(x) = y.
def: bijective
both injective and surjective
def: group
We say that a nonempty set G is a group under ⋆
if:
Closure: ⋆ is a binary operation, so g ⋆ h ∈ G for all g, h ∈ G;
Associativity: g ⋆ (h ⋆ k) = (g ⋆ h) ⋆ k for all g, h, k ∈ G;
Identity: There exists an identity element e ∈ G such that e ⋆ g = g ⋆ e = g for all g ∈ G;
Inverses: Every element g ∈ G has an inverse g^(−1)
such that g ⋆ g−1 = g^(−1) ⋆ g = e.
def: la grange’s thereom
H contains an element x of order p,
and K contains an element y of order q.
def: basis
both linearly independent + spanning
def: subspace
A subspace of a vector space V is a subset U ⊆ V
which:
1. 0 ∈ U;
2. u + v ∈ U for all u, v ∈ U ;
3. α · u ∈ U for all α ∈ F, u ∈ U.
where addition and scalar multiplication are as defined in V .
def: dimension
the size of the basis
def: order of a group
the smallest +ve integers s.t σ^n = id
def: sgn of a group
(-1)^m where m = no. of transpositions written as a product
def: subgroup test
H ⊆ G is a subset of G if and only if
- H is not empty;
- If h, k ∈ H then h ⋆ k ∈ H;
- If h ∈ H then h^(−1) ∈ H.
