Wednesday, February 1, 2023

# What Does Relationship Mean In Math Terms

## What Does Relation Mean In Math Need An Intuitive Understanding

What Does an Inverse Relationship Mean in Math?

These two paragraphs are from wikipedia,

“In mathematics, a relation is used to describe certain properties of things. That way, certain things may be connected in some way; this is called a relation. It is clear, that things are either related, or they are not, there is no in-between. Relations are classified into four types based on mapping of elements.”

“Formally, a relation is a set of n-tuples of equal degree. Thus a binary relation is a set of pairs, a ternary relation a set of triples, and so forth. In the language of set theory, a relation between two sets is a subset of their Cartesian product.”

I understand the formal definition but i want to get an intuitive understanding of the concept so i want to know that in first paragraph what did author mean by saying “properties” ? I mean what kind of properties whose properties ?

• $\begingroup$I think the first paragraph is a little unclear. A great example of a relation on the integers is “divides”.$\endgroup$

Properties are just anything about an object of which you can say that it is right or wrong. For example, “it is red” is a property that can apply e.g. top an apple: The apple may or may not be red.

Relations are properties that involve two or more objects. For example, “this apple is larger than that one over there.” Here”is larger than” is the relation, applied to the objects “this apple” and “that one over there”. It is clear that either this apple is larger than that one over there, or it is not.

## What Is A Function

A function is a relation which describes that there should be only one output for each input we can say that a special kind of relation , which follows a rule i.e., every X-value should be associated with only one y-value is called a function.

For example:

Let us also look at the definition of Domain and Range of a function.

 Domain It is a collection of the first values in the ordered pair values). Range It is a collection of the second values in the ordered pair values).

Example:

In the relation, ,

The domain is and range is .

Note: Dont consider duplicates while writing the domain and range and also write it in increasing order.

## Logarithmic Arithmetic Is Not Normal

Youve studied logs before, and they were strange beasts. Howd they turn multiplication into addition? Division into subtraction? Lets see.

What is $\ln$? Intuitively, the question is: How long do I wait to get 1x my current amount?

Zero. Zip. Nada. Youre already at 1x your current amount! It doesnt take any time to grow from 1 to 1.

• $\ln = 0$

Ok, how about a fractional value? How long to get 1/2 my current amount? Assuming you are growing continuously at 100%, we know that $\ln$ is the amount of time to double. If we reverse it wed have half of our current value.

• $\ln = \ln = -.693$

Makes sense, right? If we go backwards .693 units wed have half our current amount. In general, you can flip the fraction and take the negative: $\ln = \ln = -1.09$. This means if we go back 1.09 units of time, wed have a third of what we have now.

Ok, how about the natural log of a negative number? How much time does it take to grow your bacteria colony from 1 to -3?

Its impossible! You cant have a negative amount of bacteria, can you? At most you can have zero, but theres no way to have a negative amount of the little critters. Negative bacteria just doesnt make sense.

• $\ln = \text$

Undefined just means there is no amount of time you can wait to get a negative amount.

## Basic Mathematical Symbols With Name Meaning And Examples

The basic mathematical symbols used in Maths help us to work with mathematical concepts in a theoretical manner. In simple words, without symbols, we cannot do maths. The mathematical signs and symbols are considered as representative of the value. The basic symbols in maths are used to express mathematical thoughts. The relationship between the sign and the value refers to the fundamental need of mathematics. With the help of symbols, certain concepts and ideas are clearly explained. Here is a list of commonly used mathematical symbols with names and meanings. Also, an example is provided to understand the usage of mathematical symbols.

Symbol

## Definition Of An Equivalence Relation

In mathematics, as in real life, it is often convenient to think of two different things as being essentially the same. For example, when you go to a store to buy a cold soft drink, the cans of soft drinks in the cooler are often sorted by brand and type of soft drink. The Coca Colas are grouped together, the Pepsi Colas are grouped together, the Dr. Peppers are grouped together, and so on. When we choose a particular can of one type of soft drink, we are assuming that all the cans are essentially the same. Even though the specific cans of one type of soft drink are physically different, it makes no difference which can we choose. In doing this, we are saying that the cans of one type of soft drink are equivalent, and we are using the mathematical notion of an equivalence relation.

An equivalence relation on a set is a relation with a certain combination of properties that allow us to sort the elements of the set into certain classes. In this section, we will focus on the properties that define an equivalence relation, and in the next section, we will see how these properties allow us to sort or partition the elements of the set into certain classes.

##### Definition: equivalence relation

Let \ be a nonempty set. A relation \ on the set \ is an equivalence relation provided that \ is reflexive, symmetric, and transitive. For \, if \ is an equivalence relation on \ and \ \ \, we say that \ is equivalent to \.

##### Example 7.8: A Relation that Is Not an Equivalence Relation

Solution

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## Definition Of Symmetric Relation

I know that the relation is symmetric if $\forall x \forall y \ xRy \implies yRx$.

Consider the set $A = \\}$and $R = \\}.$

My textbook claims that this relation is symmetric. But what about $$and$$ that are not part of the $R$ set? In the definition, it said for all $x$ and $y$, so shouldn’t this violate the symmetricity?

• 2$\begingroup$There is a typo in your first line. $\endgroup$Nov 21 ’17 at 9:29

If $$is in the relation, then we check whether$$ is in the set.

However, in this case, $$is not in R, hence we should not expect$$ to be in the relation for it to be symmetric.

• $\begingroup$Oh of course. So the interpretation is for all x and y that are in that R set. It doesn’t necessarily require the relationship to be between all of the elements in set A?$\endgroup$Nov 21 ’17 at 9:41
• 1;Siong Thye GohNov 21 ’17 at 9:42
• $\begingroup$Thank you! I am so amazed how quickly the community here helps you out. This is great :)$\endgroup$

Definition 1: A relation R over set A is symmetric if for all x, y from A the following is true: is in R implies is in R.

Definition 2: A relation R over set A is symmetric if for all x, y from A the following is true: if is in R, then is in R.

These definitions do not require that every has to be in R. They only require that for those that are in R, has to be in R.

Suppose that no single , where x is different from y, is in R. In this case R is symmetric.

This is so because proposition

  if  is in R, then   is in R
   is in R

## Definition Of Totality In Relations

I see two apparently different definitions for totality which don’t seem to be equivalent.

Definition 1. A relation $R \subset X \times Y$ is total if it associates to every $x \in X$ at least one $y \in Y$; that is

$\forall x \in X \;\; \exists y \in Y: \in R$.

References:

• Gunther Schmidt and Thomas Strohlein, Relations and Groups, page 54, Springer.
• Chris Brink et al, Relational Methods for Computer Science, page 5, Springer.
• Definition 1′. A relation $R$ over $X$ is total if

$\forall a, b \in X: \in R \vee \in R$.

Reference:

Can you please elaborate on the incompatibility of these two definitions and the reasons for it? Is there any de facto standard? Would you suggest other names ?

Looking at a number of resources on the Web, it seems that left-totality is a more common name for the first definition. For example see:

Totality seems to be a more common name for the second definition. I would personally prefer “strictly connected” for the second definition because if you call it totality, one would expect to see a connection between that and left-totality or right-totality whereas I don’t think there is one.

They do not seem incompatible to me since they talk about different types of ‘totality’. The first definition takes 2 sets $X, Y$. While the second definition uses only one set.

You could transform the first definition so that it uses one set:

Roughly said:

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## Using Natural Logs With Any Rate

Sure, you say, This log stuff works for 100% growth but what about the 5% I normally get?

Its no problem. The time we get back from $\ln$ is actually a combination of rate and time, the x from our $e^x$ equation. We just assume 100% to make it simple, but we can use other numbers.

Suppose we want 30x growth: plug in $\ln$ and get 3.4. This means:

• $e^x = \text$
• $e^ = 30$

And intuitively this equation means 100% return for 3.4 years is 30x growth. We can consider the equation to be:

We can modify rate and time, as long as rate * time = 3.4. For example, suppose we want 30x growth how long do we wait assuming 5% return?

• $\ln = 3.4$
• $\text * \text = 3.4$
• $.05 * \text = 3.4$
• $\text = 3.4 / .05 = 68 \text$

Intuitively, I think “$\ln = 3.4$, so at 100% growth it will take 3.4 years. If I double the rate of growth, I halve the time needed.”

• 100% for 3.4 years = 1.0 * 3.4 = 3.4
• 200% for 1.7 years = 2.0 * 1.7 = 3.4
• 50% for 6.8 years = 0.5 * 6.8 = 3.4
• 5% for 68 years = .05 * 68 = 3.4

Cool, eh? The natural log can be used with any interest rate or time as long as their product is the same. You can wiggle the variables all you want.

## Princeton’s Wordnetrate This Definition:

What Does Diagonal Mean in Math? : Math Definitions & More
• relationnoun

an abstraction belonging to or characteristic of two entities or parts together

• sexual intercourse, intercourse, sex act, copulation, coitus, coition, sexual congress, congress, sexual relation, relation, carnal knowledgenoun

the act of sexual procreation between a man and a woman; the man’s penis is inserted into the woman’s vagina and excited until orgasm and ejaculation occur

• relative, relationnoun

a person related by blood or marriage

“police are searching for relatives of the deceased”; “he has distant relations back in New Jersey”

• relation, telling, recountingnoun

an act of narration

“he was the hero according to his own relation”; “his endless recounting of the incident eventually became unbearable”

• relation back, relationnoun

the principle that an act done at a later time is deemed by law to have occurred at an earlier time

“his attorney argued for the relation back of the amended complaint to the time the initial complaint was filed”

• relationnoun

mutual dealings or connections among persons or groups

“international relations”

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## Relations And Functions Explanation & Examples

Functions and relations are one the most important topics in Algebra. On most occasions, many people tend to confuse the meaning of these two terms.;

In this article, we will define and elaborate on how you can identify if a relation is a function. Before we go deeper, lets look at a brief history of functions.

The concept of function was brought to light by mathematicians in the 17th century. In 1637, a mathematician and the first modern philosopher, Rene Descartes, talked about many mathematical relationships in his book Geometry. Still, the term function was officially first used by German mathematician Gottfried Wilhelm Leibniz after about fifty years. He invented a notation y = x to denote a function, dy/dx, to denote a functions derivative. The notation y = f was introduced by a Swiss mathematician Leonhard Euler in 1734.

Lets now review some key concepts as used in functions and relations.

• What is a set?

A set is a collection of distinct or well-defined members or elements. In mathematics, members of a set are written within curly braces or brackets . Members of assets can be anything such as; numbers, people, or alphabetical letters, etc.

For example,

is a set of alphabet letters.

is a set of even numbers.

is a set of prime numbers

Two sets are said to be equal; they contain the same members. Consider two sets, A = and B = . Regardless of the members position in sets A and B, the two sets are equal because they contain similar members.

• What is a domain?

## What Does A Linear Relationship Tell You

There are three sets of necessary criteria an equation has to meet in order to qualify as a linear one: an equation expressing a linear relationship can’t consist of more than two variables, all of the variables in an equation must be to the first power, and the equation must graph as a straight line.

A commonly used linear relationship is a correlation, which describes how close to linear fashion one variable changes as related to changes in another variable.

In econometrics, linear regression is an often-used method of generating linear relationships to explain various phenomena. It is commonly used in extrapolating events from the past to make forecasts for the future. Not all relationships are linear, however. Some data describe relationships that are curved while still other data cannot be parameterized.

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## Special Types Of Binary Relations

Some important types of binary relations R over sets X and Y are listed below.

Uniqueness properties:

• Injective : for all x, z â X and all y â Y, if xRy and zRy then x = z. For such a relation, is called a primary key of R. For example, the green and blue binary relations in the diagram are injective, but the red one is not , nor the black one .
• Functional #cite_note-kkm-17″ rel=”nofollow”>right-definite or univalent): for all x â X and all y, z â Y, if xRy and xRz then y = z. Such a binary relation is called a partial function. For such a relation, is called a primary key of R. For example, the red and green binary relations in the diagram are functional, but the blue one is not , nor the black one .
• One-to-one: injective and functional. For example, the green binary relation in the diagram is one-to-one, but the red, blue and black ones are not.
• One-to-many: injective and not functional. For example, the blue binary relation in the diagram is one-to-many, but the red, green and black ones are not.
• Many-to-one: functional and not injective. For example, the red binary relation in the diagram is many-to-one, but the green, blue and black ones are not.
• Many-to-many: not injective nor functional. For example, the black binary relation in the diagram is many-to-many, but the red, green and blue ones are not.

Totality properties :

Uniqueness and totality properties :

## Representation Of Relation In Math:

The relation in math from set A to set B is expressed in different forms.;;;; Roster form;

;;; Arrow diagram;

i. Roster form:;

In this, the relation from set A to B is represented as a set of ordered pairs.

In each ordered pair 1st component is from A; 2nd component is from B.

Keep in mind the relation we are dealing with. For Example: 1. If A = B =

then R =

Hence, R A × B

2. Given A = B = then the relation R from A to B is defined as is less than and can be represented in the roster form as R =

Here, 1 component < 2 component. In roster form, the relation is represented by the set of all ordered pairs belonging to R. If A = and B =

if a R b means a² = b

then, R = {, ,

ii. Set builder form: In this form, the relation R from set A to set B is represented as R = , the blank space is replaced by the rule which associates a and b. For Example: Let A = and B =

Let R = {, , , then R in the set builder form, it can be written as

R = : a A, b B, a is 2 less than b}

iii. Arrow diagram: Draw two circles representing Set A and Set B.

Write their elements in the corresponding sets, i.e., elements of Set A in circle A and elements of Set B in circle B.

Draw arrows from A to B which satisfy the relation and indicate the ordered pairs.

2. If A = and B = and R be the relation ‘is less than’ from A to B, then R =

;Relations and Mapping

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