what is an injective function
f : A → B is called injective or one-to-one if for every x,y ∈ A, f(x) = f(y) implies x = y
what is a surjective function
- what is an interesting note to make about the image of f with surjectiveness
A function f : A → B is called surjective or onto if for every y ∈ B, there exists x ∈ A, such that f(x) = y
- the image f of A has to be all of B
If f and g are both injective, then g ◦ f is what
injective
explain how cycle notation works for permutations
- If a1, a2, . . . , ak are distinct elements of {1, 2, . . . , n}, we write (a1 a2 a3 . . . ak)
for the permutation f defined by:
f(a1) = a2
f(a2) = a3
f(a3) = a4
.
.
.
f(ak−1) = ak
f(ak) = a1 and
f(j) = j for all j not in the set {a1, a2, . . . , ak}.
We call (a1 a2 . . . ak) a k-cycle (or just a cycle).
what is a permutation
A bijective function f: {1, 2, …., n} → {1, 2, …., n}
for which values of n is Sn not commutative
n ≥ 3
so what type of cycles are commutative
disjoint
what is the well-ordering principle
Every non-empty subset of Z that is bounded below has a least element
. Let a, b be integers that are not both zero, and let g = gcd(a, b). Then if d|a and d|b,
d|g
. Let a, b, c ∈ Z. Assume a and b are relatively prime.
1. If a|bc,
a|c
Assume a and b are relatively prime
If a|c and b|c
then ab|c
If c|a and c|b, then ???
c|(ax + by) for any x, y ∈ Z
If a|b and b|c ???
then a|c
