Relations and Inverse Relations

In summary, the conversation discusses a problem involving a relation ##R## defined on ##N##, where ##aRb## if ##\frac{a}{b} \in N##. The exercise asks for the conditions under which ##c R^{-1} d## holds, where ##R^{-1}## is the inverse relation of ##R##. From the definitions provided, it is concluded that ##c R^{-1} d## holds if and only if ##\frac{d}{c} \in N##.
  • #1
Tsunoyukami
215
11
I am having difficulty understanding the following problem. I feel it should be very simple but am unsure how to interpret it.

A relation ##R## is defined on ##N## by ##aRb## if ##\frac{a}{b} \in N##. For ##c, d \in N##, under what conditions is ##c R^{-1} d##? (Exercise 8.6 from Chartrand, Polemni & Zhang's Mathematical Proofs: A Transition to Advanced Mathematics; 3rd edition; page 210).


From page 192 of the same text, "Let ##A## and ##B## be two sets. By a relation ##R## from ##A## to ##B## we mean a subset of ##A x B##. That is, ##R## is a set of ordered pairs, where the first coordinate of the pair belongs to ##A## and the second coordinate belongs to ##B##. If ##(a,b) \in R##, then we say that ##a## is related to ##b## by ##R## and write ##a R b##. If ##(a,b) \notin R##, then ##a## is not related to ##b## by ##R## and we write ##a ~R b##." (Here I have used ##~R## to denote "not ##R##" because I was unsure how to cross ##R## out as it appears in the text).

On the next page (193), "Let ##R## be a relation from ##A## to ##B##. By the inverse relation of ##R## is meant the relation ##R^{-1}## from ##B## to ##A## defined by
##R^{-1}=\left\{ (b,a) : (a,b) \in R \right\}##."


Given these definitions I am having difficulty understanding what the notation means. For example, I understand what ##a R b## means, but I do not understand what ##a R^{-1} b## means. Because of the way ##R^{-1}## is defined (that is, ##R^{-1}=\left\{ (b,a) : (a,b) \in R \right\}##) I am not sure whether this definition is precisely ##a R^{-1} b## or ##b R^{-1} a##.

Any help with regard to this would be much appreciated. Thank you in advance!
 
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  • #2
Basically, we have that ##aR^{-1}b## if and only if ##bRa##.
 
  • #3
So then the solution to the problem is:

Because ##c R^{-1} d## if ##d R C##, and by the way R is defined, ##d R c## if ##\frac{d}{c} \in N##. So the condition for ##c R^{-1} d## is that ##\frac{d}{c} \in N##.
 
  • #5
In particular,and this may be the point of the exercise, [itex]aRb[/itex] and [itex]aR^{-1}b[/itex] if and only if a/b= 1 or -1.
 

Related to Relations and Inverse Relations

What is a relation?

A relation is a set of ordered pairs that relate two sets of numbers or objects to each other. Each ordered pair consists of an input value and an output value, showing how the two sets are related.

What is an inverse relation?

An inverse relation is a relation in which the input and output values are switched. In other words, the roles of the independent and dependent variables are reversed. This means that for every output value in the original relation, there is a corresponding input value in the inverse relation.

How can you tell if a relation is a function?

A relation is a function if each input value has only one output value. This means that each input value is mapped to exactly one output value in the relation. To determine if a relation is a function, you can use the vertical line test. If a vertical line can be drawn through the graph of the relation and only intersects it at one point, then the relation is a function.

What is the domain of a relation?

The domain of a relation is the set of all input values for which the relation is defined. In other words, it is the set of all values that can be used as the independent variable in the relation. The domain is typically represented as the set of x-values in the relation's ordered pairs.

What is the range of a relation?

The range of a relation is the set of all output values that are produced by the relation. In other words, it is the set of all values that can be used as the dependent variable in the relation. The range is typically represented as the set of y-values in the relation's ordered pairs.

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