# Tagged Questions

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### 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 $m|n$.
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### 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 ...
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### 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 ...
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### 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 ...
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### 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: ...
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### 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 ...
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### 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 ...
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### 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 $$f(x)=\frac{9}{9-x^2}+\log_{10}(x^3-x)$$ ...
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### 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 ...
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### 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 $arb$ ...
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### 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 $arb$ ...
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### 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 $\emptyset$? ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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: $f[f(x)]=g[f(x)]$ ...
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### 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?
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### 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?
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### 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?
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### 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. ...
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### 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 ...
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### 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? ...
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### 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 ...
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### 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 ...
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### 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 $f(f(x))=x$ ...
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### 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 ...
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### 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 ...
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### 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?
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### 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 ...
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### 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 ...
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### $M_1 = (x,y)\quad x²+y²+6y = 7$ 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 ...
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### 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 ...
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### 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 ...
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### 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 $$R := \bigl\{\left(a,b\right) \mid \text{a,b \in A and f(a)=f(b)}\bigr\}.$$ ...
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### 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 ...
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### 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.
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### 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 ...
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### 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 ...
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### 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 $f(m+n)$.
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### 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 $=$ ...