Let F be a finite field. Prove that there exists an integer n≥1,
such that n.1_F = 0_F .
Show further that the smallest positive integer with this
property is a prime number.
1. Prove or disprove: if f : R → R is injective and g : R → R is
surjective then f ◦ g : R → R is bijective.
2. Suppose n and k are two positive integers. Pick a uniformly
random lattice path from (0, 0) to (n, k). What is the probability
that the first step is ‘up’?
Some hints: use the definition: f is a function iff a = b
implies f(a) = f(b) and recall that in informal proofs we show an
implication by assuming the if part of the implication, and then
deducing the then part of the implication.
The base case will show that a = b implies f(a) = f(b) when f(x)
= c0 (a constant function). The inductive case will
assume a = b implies f(a) = f(b) for degree k, and...
Classify each function as injective, surjective, bijective, or
none of these.
a) f1 : N --> Q defined by f1 (n) =
n/(n+1)
b) f2: Z --> Z defined by f2(n) =
n2
c) f3 : N --> N defined by f3(n) =
n3
d) f4 : B --> (0, infinity) defined by
f4 (C) = the area of C, where B is the set of all
circles in the firs quadrant that are simultaneously tangent to
both the positive...
Prove that there exists integers m and n such that 15m + 12n =
3
Please do not prove by assuming m=1 and n=-1, I'd like to prove
by not assuming any actual numbers.