Tagged Questions

3
votes
1answer
25 views

Preimages of a function: Is the following proposition true or false?

Let $g: ℤ \times ℤ → ℤ \times ℤ$ be defined by $g(m,n) = (2m, m – n)$. Is the following proposition true or false? Justify your conclusion. For each $(s, t) ∈ ℤ \times ℤ$, there exists an $(m, n) ∈ ℤ ...
0
votes
1answer
30 views

prove $f^{-1}(B)=A$

I am given $A_1$, $A_2 \subseteq A$ and $B_1$,$B_2 \subseteq B$. and the function $f: A \rightarrow B$ I want to prove that $f^{-1}(B)=A$. I just assume that here one is talking about ...
5
votes
2answers
70 views

How to exactly write down a proof formally (or how to bring the things I know together)?

I've always had trouble with this: I know everything I need to know but I can't connect everything I know in the way it's being wanted from me. This is what I have to do: Prove for $f : M → N$ the ...
1
vote
1answer
39 views

Proving $\forall f\in \mathscr F\exists g\in \mathscr F $ so that $g(f(1)) = 2$.

Let $\mathscr F$ denote the set of all functions from $\{1,2,3\}$ to $\{1,2,3\}$. Prove or disprove. (a) $\forall f\in \mathscr F\exists g\in \mathscr F $ so that $g(f(1)) = 2$. (b) $\forall f\in ...
0
votes
0answers
32 views

I wanna prove if the composite are equal to each other

Given $f : \{0,1\}^n \to \{0,1\}^n$, define $f': \{0,1\}^{2n} \to \{0,1\}^{2n}$ as follows: for $x, r \in \{0,1\}^n$ define $f'(x \circ r) := f(x) \circ r$ (where $\circ$ denotes concatenation). Prove ...
8
votes
8answers
508 views

How do I prove that a function is well defined?

How do you in general prove that a function is well-defined? $$f:X\to Y:x\mapsto f(x)$$ I learned that I need to prove that every point has exactly one image. Does that mean that I need to prove the ...
1
vote
1answer
59 views

Special case of combinatorial onto functions

Let $[n]$ be the set of integers $\{1, 2, \ldots, n\}$. I want to find the number of onto functions from $[m]$ to $[3]$. The answer I found was $3!\cdot (3)^{m-3}$ My reasoning is: We have a ...
1
vote
1answer
44 views

Combinatorial Correctness of one-to-one functions

Let $\lbrack k\rbrack$ be the set of integers $\{1, 2, \ldots, k\}$. What is the number of one-to-one functions from $m$ to $n$ if $m \leq n$? My answer is: $\dfrac{n!}{(n-m)!}$ My reasoning is the ...
4
votes
1answer
113 views

prove $x \mapsto x^2$ is continuous

I am to show the continuity of this function with the help of $\epsilon$-$\delta$ argument. The function is: $g: \Bbb{R} \rightarrow \Bbb{R}$, $x \mapsto x^2$. Given the $\epsilon$-$\delta$ ...
2
votes
2answers
102 views

If $f: A \rightarrow B$ is injective, $\exists ! f^{-1}:f(A)\rightarrow A$ such that $f^{-1}f(x) = x$

If $f: A \rightarrow B$ is injective, $\exists ! f^{-1}:f(A)\rightarrow A$ such that $f^{-1}f(x) = x$ I know the proof of this is super simple, like 2 lines (+ some for uniqueness), but I can't ...
0
votes
2answers
58 views

Maps - question about $f(A \cup B)=f(A) \cup f(B)$ and $ f(A \cap B)=f(A) \cap f(B)$

I am struggling to prove this map statement on sets. The statement is: Let $f:X \rightarrow Y$ be a map. i) $\forall_{A,B \subset X}: f(A \cup B)=f(A) \cup f(B)$ ii) $\forall_{A,B \subset X}: ...
3
votes
1answer
354 views

If $F$ and $G$ are one to one, then $G \circ F$ is one to one and $(G \circ F)^\neg = F^\neg \circ G^\neg$

THEOREM: if $F$ and $G$ are one to one then $G \circ F$ is also one to one and $(G \circ F)^\neg$ = $F^\neg \circ G^\neg$ PROOF: if $F: A\rightarrow B$, $G: B \rightarrow C$ and $$\forall a, a' ...
4
votes
1answer
73 views

If $\operatorname{ran} F \subseteq \operatorname{dom} G$, then $\operatorname{dom}(G \circ F) = \operatorname{dom} F$.

THEOREM: If $ \text{ran } F \subseteq \text{dom } G $ then $\text{dom }(G \circ F)= \text{dom }F$ PROOF: if $ F\subseteq A \times B$ and $ G\subseteq B\times C$ then by definition ...
0
votes
2answers
324 views

Invertibility of a function and left/right inverses

I am new to writing proofs, as a result even when i may know an answer i sometimes doubt if i know how to write the proof. So here is the problem which should be an easy one. In fact i think the proof ...
1
vote
0answers
63 views

Proving That The Graph Is Symmetric About The Origin

I know that the graph of an equation in x and y is symmetric with respect to the origin if replacing x by -x and y by -y yeilds the same equation. How can one prove that if the graph is symmetric ...
-3
votes
3answers
2k views

Proving a function is one-to-one

what is the easiest way to prove a function si one to one? Also to prove it's not one to one... Define $f : \mathbb{N} \to \mathbb{N}$ by $f(n) = \cdots$. Define $g : \mathbb{N}\times\mathbb{N} ...
1
vote
1answer
695 views

Proofs of Hyperbolic Functions

I know that functions which are associated with the geometry of the conic section called a hyperbola are called hyperbolic functions. But where on earth did '$e$' come from? I really don't ...
5
votes
2answers
153 views

Need help with very simple set theoretic proofs

I am self studying Munkres' Topology book, and I'm having a hard time writing down proofs that relate to set theory. I can see why certain arguments are true, but constructing a formal proof seems to ...
0
votes
2answers
100 views

how can I present an idea about the difference between two functions clearly?

I have a presentation in which I want to point out a difference between two functions. Instead of putting the two functions on a slide and pointing to the differences, I want to do something simple. ...