Question

Let f be a function from the set of students in a discrete mathematics class to the set of all possible final grades.

(a) Under what conditions is f an injection?

(b) Under what conditions is f a surjection?

(please show all work)

Answer #1

Let A be a finite set and let f be a surjection from A to
itself. Show that f is an injection.
Use Theorem 1, 2 and corollary 1.
Theorem 1 : Let B be a finite set and let f be a function on B.
Then f has a right inverse. In other words, there is a function g:
A->B, where A=f[B], such that for each x in A, we have f(g(x)) =
x.
Theorem 2: A right inverse...

Discrete mathematics function relation
problem
Let P ∗ (N) be the set of all nonempty subsets of N. Define m :
P ∗ (N) → N by m(A) = the smallest member of A. So for example, m
{3, 5, 10} = 3 and m {n | n is prime } = 2.
(a) Prove that m is not one-to-one.
(b) Prove that m is onto.

Let g be a function from set A to set B and f be a function from
set B to set C. Assume that f °g is one-to-one and function f is
one-to-one. Using proof by contradiction, prove that function g
must also be one-to-one (in all cases).

Let X be the set {1, 2, 3}.
a)For each function f in the set of functions from X to X,
consider the relation that is the symmetric closure of the function
f'. Let us call the set of these symmetric closures Y. List at
least two elements of Y.
b) Suppose R is some partial order on X. What is the smallest
possible cardinality R could have? What is the largest?

Let A be a finite set and f a function from A to A.
Prove That f is one-to-one if and only if f is onto.

Discrete Mathematics
(a) Let P(x) be the predicate “−10 < x < 10” with domain
Z+ (the set of all positive integers). Find the truth set of
P(x).
(b) Rewrite the statement Everybody trusts somebody in formal
language using the quantiﬁers ∀ and ∃, the variables x and y, and a
predicate P(x,y) that you must deﬁne.
(c) Write the negation of the statement in (b) both formally and
informally.

discrete mathematics
1. How many element are in A1 È A2 if there are 12 elements A1,
18 elements in A2 and
a) A1 Ç A2 =Æ?
b) | A1 Ç A2 | =1?
c) | A1 Ç A2 | =6?
d) A1 Í A2?
2. There are 345 students at a college who have taken a course
in calculus, 212 who have taken a course in discrete mathematics
and 188 who have taken courses in both calculus and discrete...

Let f be a function which maps from the quaternion group, Q, to
itself by f (x) = i ∙x, for i∈ Q and each element x in Q. Show all
work and explain! (i) Is ? a homomorphism? (ii) Is ? a 1-1
function? (iii) Does ? map onto Q?

Let (X, d) be a metric space, and let U denote the set of all
uniformly continuous functions from X into R. (a) If f,g ∈ U and we
define (f + g) : X → R by (f + g)(x) = f(x) + g(x) for all x in X,
show that f+g∈U. In words,U is a vector space over R. (b)If f,g∈U
and we define (fg) : X → R by (fg)(x) = f(x)g(x) for all x in X,...

Let X be a set and A a σ-algebra of subsets of X.
(a) A function f : X → R is measurable if the set {x ∈ X : f(x)
> λ} belongs to A for every real number λ. Show that this holds
if and only if the set {x ∈ X : f(x) ≥ λ} belongs to A for every λ
∈ R. (b) Let f : X → R be a function.
(i) Show that if...

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