0
votes
2answers
19 views

Relations on set A and conditions of existence of descending chains

I am reading through the Elements of Set theory by Herbert Enderton and even though I have passed this exercise long ago,the more I look at my solution the less I believe it. Problem goes like this: ...
8
votes
1answer
612 views

What is meant by “m|n”? Two letters separated by a vertical bar (|)

I am new to this subject, and not not sure what "|" symbol means on this statement. Let $R_2 \subset\Bbb N \times\Bbb N$ be defined by $(m, n) \in R_2$ if and only if $m|n$.
2
votes
1answer
40 views

Relations and functions with valence 0

From http://en.wikipedia.org/wiki/First-order_logic#Non-logical_symbols Relations of valence 0 can be identified with propositional variables. For example, P, which can stand for any ...
0
votes
3answers
61 views

What would make a function reflexive, transitive, and/or symmetric?

A binary relation $R$ is a subset of the Cartesian product between two sets $X$ and $Y$, containing a set of ordered pairs $\{(x,y) : x \in X, y \in Y\}$. $R$ is a function if each element of $X$ is ...
0
votes
1answer
40 views

Understanding the difference between relations and functions.

$R=\{(1,2),(1,3)\}$ is a relation but not function. The logic for this is that if the first element of every ordered pair must remain different, then it is said to be function. Otherwise, it's just ...
0
votes
0answers
10 views

For which sets, $X$ the relation is a partial function

Given $T=\left\{\ \left<A,B\right> \in (P(X))^2 | A\subseteq B \right\}$ For which sets, $X$, the relation $(P(X))^2-T \cap (P(X))^2-T^{-1}$ is a partial function? Here's my solution: ...
2
votes
0answers
60 views

Using the ELO Rating System on Static Objects

The Setup Suppose we have a list of movies $m_1, m_2, \dots, m_n$ that we wish to rank in order of "quality." We define the "strength" of a movie $a$ by a function $f$ which takes in numerical ...
1
vote
2answers
49 views

Find the $f(x)$ from the given information

So tomorrow I tackled a maths test where I faced a question which was saying, Question: Let $f:R-\{0,1\}\rightarrow R$ be a function satisfying the relation ...
1
vote
1answer
30 views

Domain of definition of the function

I was going through some questions of Relations and Functions and now I am stuck to one. Question says Question: Domain of definition of the function $$f(x)=\frac{9}{9-x^2}+\log_{10}(x^3-x)$$ ...
0
votes
1answer
17 views

What's the difference between a partial function and a relation?

My understanding of a partial function is that it is one which only maps a subset of some set $A$ to another set $B$ (where $B$ could be $A$). On the Wikipedia page, the below image is given as an ...
1
vote
1answer
11 views

Value assignment for complete, transitive relation on uncountably infinite set

Consider a set $A$ and a relation $r$. The relation $r$ is complete, i.e., for any $a,b\in A$, we have $arb$ or $bra$ or both. The relation $r$ is transitive, i.e., for any $a,b,c\in A$, if $arb$ ...
2
votes
1answer
21 views

Value assignment for complete, transitive relation on countably infinite set

Consider a set $A$ and a relation $r$. The relation $r$ is complete, i.e., for any $a,b\in A$, we have $arb$ or $bra$ or both. The relation $r$ is transitive, i.e., for any $a,b,c\in A$, if $arb$ ...
0
votes
1answer
26 views

Inverse image of an element in co-domain but not in range?

Sorry, quite new to this. I have a question that contains the image below of $g:X\rightarrow Y$ and it is asking for the inverse image of $u$. Am I correct in thinking that the answer is $\emptyset$? ...
3
votes
3answers
73 views

Find $f(2a-x)$ from given equation

I have been trying to solve an exercise based on the relations and functions. Right now I had stuck to a question based on functions. The question says: A real valued function $f(x)$ satisfies the ...
0
votes
1answer
19 views

Prove that $P(\Bbb R)/\prec$ is countable and show that the class $[A]_\prec$ is an infinite set and not countable

Let $\prec$ be the relation over $P(\Bbb R)$ defined as: $A \prec B$ if and only if $|A \cap \Bbb Q| = |B \cap \Bbb Q|$. Prove that the quotient set $P(\Bbb R)/\prec$ is countable and show that ...
0
votes
2answers
27 views

Bijection of a function.

Define the function f: $(2,\infty) -> (-\infty,-1)$ by $f(x)= \frac{-x}{x-2}$. Show that f is bijective. I know i need to prove both injective and surjective, and I was able to solve the equation ...
1
vote
2answers
36 views

Find the value of $x$ for which $ff=gf$.

Functions $f$ and $g$ are defined by $f:x \mapsto \frac{1}{2x+1}$, $x \neq \frac{-1}{2}$ and $g:x \mapsto x+1$. Find the value of $x$ for which $ff=gf$. So I started in this way: $f[f(x)]=g[f(x)]$ ...
1
vote
1answer
41 views

Find the fuction $g$.

If $f:x \mapsto x^2 + 3$, find function $g$ such that $gf:x \mapsto 2x^2 + 3$. I don't know how to do it, there is no such example in my book. Help?
0
votes
1answer
25 views

State the range of the function below.

Sketch the graph of $f:x \mapsto -4x + 5$ , $x<2$ and state the range. I got the graph, but can't state the range...how to find them?
0
votes
0answers
64 views

Draw arrow diagram to show the following function.

Draw arrow diagram with two parallel lines to show the function $f:x \mapsto 3 - 2x^2$. Let the domain be the set of integers and draw six arrows for the function. How to draw it?
0
votes
1answer
44 views

Equivalence Relation on R (real numbers)

Let R be the relation on R(real numbers) defined by: For all x, y (that belong) to R(real numbers), x relates y <=> x-y (that belongs) to Z. (a) Is R an equivalence relation? Prove your answer. ...
1
vote
1answer
59 views

Let A = {1,2,3,4} Let F be the set of all functions from A to A. (check the parts)

Let $\operatorname{S}$ be a relation on $F$ defined by: $\forall f, g \in F, f\,\operatorname{S}\,g \iff f(i) = g(i), \exists i \in A$. (a) Recall that the identity function $I_A : A \mapsto A$ is ...
3
votes
1answer
47 views

Discrete math functions help?

I'm doing a review for my discrete math test on functions and I'm having troubles with a few questions. Can I get some guidance in how to do these questions so I can be more prepared for the test? ...
1
vote
2answers
40 views

Transitive & symmetric relation; why is this wrong?

"Give a relation that satifies the condition:" Symmetric and transitive but not reflexive. This is what I gave: R = {(x,y), (y,z), (z,x), (y,x), (z,y), (x,z)} I was told this was not ...
1
vote
1answer
830 views

Prove that between every two rational numbers a/b and c/d that there is a rational number and there are an infinite number of rational numbers [duplicate]

So the full problem is Prove that between every two rational numbers $a/b$ and $c/d$ that: There is a rational number There are an infinite number of rational numbers I am having ...
1
vote
1answer
25 views

Show that f is a symmetric relation on A

I am learning about relations and I come across this exercise. And I don't understand the problem. Let me first state the problem here: Let $f: A \rightarrow A$ be a function for which $f(f(x))=x$ ...
1
vote
1answer
57 views

On the size of a set of functions such that $f(i)\ne f(i+1)$ for every $i$ (and similar conditions)

For a finite set $A$,let $|A|$ denote the number of elements in the set $A$. (a) Let $F$ be the set of all functions $$f: \{1,2,\ldots,n \} \to \{1,2,\ldots,k\}~~~~~~~~~~ (n\ge 3,k\ge 2)$$satisfying ...
1
vote
3answers
72 views

Given 2 sets (X and Y) is it possible for $f: Y \to X $ to be a relation, or not?

This question is from my Computational Theory course's homework. I completely understand functions and relations (I've taken numerous Calculus courses). Here's a general example of what the question ...
0
votes
2answers
52 views

Relations between two functions

Consider the statements (1) "If $f(i) \geq f(j)$ then $q(i) \geq q(j)$", and (2) "If $q(i) < q(j)$ then $f(i) \leq f(j)$". How can we relate these statements? I mean are these related?
0
votes
1answer
78 views

Finding the number of different relations and functions

This must be a very stupid question. Let set $A=\lbrace{a,b\rbrace}$ and $B=\lbrace{1,2,3\rbrace}$. The total number of relations from $A$ to $B$ is $6$. We can calculate this as a has $3$ choices and ...
8
votes
3answers
163 views

Can we take images of equivalence relations?

Given a function $f : X \rightarrow Y$, it is well-known that we can take the image under $f$ of any subset $A \subseteq X$, and we can take the preimage under $f$ of any subset $A \subseteq Y$. This ...
0
votes
1answer
36 views

$M_1 = (x,y)\quad x²+y²+6y = 7 $ to $x \rightarrow y$

I have two relations: $$M_1 = (x,y)\qquad x²+y²+6y = 7 $$ $$M_2 = (x,y)\qquad x²+y²-6x = 7, \qquad y \ge 0$$ The question is if this relations also reflex functions like $x \rightarrow y$? I ...
2
votes
2answers
48 views

How to find inverse of function $f(x, y)$?

I am aware of the method to find inverse function $f^{-1}(x)$ of $f(x)$, which is Replace $f(x)$ with $y$ Switch $x$'s and $y$'s Solve for $y$ Replace $y$ with $f^{-1}(x)$ the above method ...
-1
votes
1answer
41 views

Induced mappings

Suppose $f$ is a mapping from the powerset of $A$ to the powerset of $B$. Let $S$ and $T$ be subsets of $A$. If both $f(\varnothing)=\varnothing$, and $f(S \cup T) = f(S) \cup f(T)$, then is $f$ the ...
-2
votes
1answer
172 views

How to prove a relation is reflexive and transitive. [closed]

In a question paper (I downloaded from internet) there was a question, Let $f\colon A\to B$ be a function. Define $$R := \bigl\{\left(a,b\right) \mid \text{$a,b \in A$ and $f(a)=f(b)$}\bigr\}.$$ ...
2
votes
2answers
47 views

Equivalence Classes of a Relation Given as a Set of Ordered Pairs

Question: The relation R is an equivalence relation on the set A. Find the distinct equivalence classes of R. A = {a, b, c, d} R = {(a, a), (b, b), (b, d), (c, c), (d, b), (d, d)} My work: So when ...
2
votes
5answers
58 views

What functions $f: A \to B$ and $g: B \to A$, satisfy a restriction such that $f$ is not invertible but $f \circ g=id_B$?

I am caught up on the notation of $id_B$. I'm thinking that $f=x^2$, or something along those lines, but not so sure as to what $g$ may be.
1
vote
4answers
199 views

is the empty set a relation?

Is the empty set is a relation? I wonder if the empty set is a relation.in enderton's a relation is a set of ordered pairs. If yes it's a relation why is that?. There is an example in the text for a ...
1
vote
2answers
117 views

Notation of a function that maps a random element

Let there be a functions $f$ and $g$ such that, $$f:A \times B \mapsto \Re$$ $$g: B \mapsto A$$ where $\forall b \in B$, $g(b)$ is some $a$ such that, $\forall a' \in A, f(a,b) \geq f(a',b)$. (This ...
0
votes
4answers
119 views

How to find $f(2013)$ if $f(5)=45$ and $f(m)+f(n)= f(m+n)$ for all $m,n\in\mathbb N$?

$f: \mathbb N\to\mathbb N$, $f(m)+f(n)=f(m+n)$ for all $m,n\in\mathbb N$, and $f(5)=45$. Find $f(2013)$. I messed up my original posting, its fixed now. I changed $m+ n$ to $f(m+n)$.
0
votes
0answers
15 views

How to work with function properties when it is defined by an equation system?

Have $A = \{1,2,3\}$, $B = \{1,2,3,4\}$, and $f:A\times A \rightarrow > B$ defined by: $$f((a,b)) = \begin{cases} 1 \textrm{ if } a < b \\ 3 \textrm{ if } a > b \\ 4 \textrm{ if } a ...
0
votes
0answers
26 views

“Anti-cumulative” Relation Image using Intersection

Given a binary relation $R \subseteq X \times Y$, the familiar image of some $A \subseteq X$ is defined as $R[A] = \{y\ |\ (x, y) \in R, x \in A\}$. Naturally we have the property $R[A] = \bigcup_{x ...
3
votes
2answers
70 views

Relations and Functions - Is my answer correct?

Could someone please advise if my answer is correct or incorrect? Any help will be greatly appreciated. Given the sets $A = \{1, 2, 3\}$, $B = \{−1, 0, 1, 2\}$ and $C = \{3, 4, 5, 6\}$, indicate the ...
0
votes
0answers
23 views

Restricting binary relations by composing with an “inclusion binary relation”

If $X' \subseteq X$ then we may define an inclusion map $\iota : X' \to X$ where $\iota(x) = x$. One use of $\iota$ is that we can express the restriction of some $f : X \to Y$ to $X'$ as $f|_{X'} = f ...
0
votes
1answer
290 views

Antisymmetric Relation

Determine whether the relation R on the set of all people is antisymmetric. (a) a is taller than b. (b) a and b are born on the same day. (c) a has the same first name as b. ...
4
votes
1answer
101 views

$F : \mathbb{Z} \to \mathbb{Z}$, $F(n) = 2 -3n$. Is $F$ one-to-one? Onto?

Define $F : \mathbb{Z} \to \mathbb{Z}$ by the rule $F(n) = 2 -3n$, for all $n \in \mathbb{Z}$. Is $F$ one-to-one? Onto? Now, I understand that one-to-one means that nothing in the co-domain is ...
1
vote
2answers
211 views

A function on binary relations

Let $\rho$ is a function mapping every binary relation $f$ (on some set $U$) into a function which maps binary relations into binary relations by the formula $$(\rho(f))(g) = f\circ g.$$ Is $\rho$: ...
0
votes
2answers
70 views

Prove $f(\cap \scr{C}) \subset \cap f(\scr{C})$. Confused on why it's not a symmetric relation?

If there are any minor mistakes in my proof, it would be great if they were pointed out - but let it not be the central discussion. I'm rather concerned why the answer is $\subset$ instead of $=$ ...
0
votes
4answers
97 views

bijection in $\mathcal P \left({S}\right)$ and $\mathcal P \left({S}\right) \times \mathcal P \left({S}\right)$

given that ${S}$ is countably infinite set. is there any bijection exist between $\mathcal P \left({S}\right)$ and $\mathcal P \left({S}\right) \times \mathcal P \left({S}\right)$. Here $\mathcal ...
0
votes
1answer
65 views

Bijection between $\mathbb{R} \times \mathbb{R}$ and $\mathbb{R}$

How can we define bijection in between $\mathbb{R} \times \mathbb{R}$ and $\mathbb{R}$? Even giving a injection from $\mathbb{R} \times \mathbb{R}$ to $\mathbb{R}$ and vice-versa will work.