# Reflexive Property Of Equality

**Reflexive Property: **If you look in a mirror, what do you see? Your reflection! You are seeing an image of yourself. You could look at the reflexive property of equality as when a number looks across an equal sign and sees a mirror image of itself! Reflexive pretty much means something relating to itself.

The **reflexive property of equality** simply states that value is equal to itself. Further, this property states that for all real numbers, *x* = *x*. What is a real number, though?

**Real numbers** include all the numbers on a number line. They include rational numbers and irrational numbers. A **rational number** is any number that can be written as a fraction. An **irrational number**, on the other hand, is a real number that cannot be written as a simple fraction. Square roots would be in this category. In fact, real numbers pretty much entail every number possible except for negative square roots because they are imaginary numbers.

Therefore, the reflexive property of equality pretty much covers most values and numbers. Again, it states simply that any value or number is equal to itself.

## What is the reflexive property in geometry?

**Reflexive Property** of Congruence. The **reflexive property** of congruence states that any **geometric** figure is congruent to itself. … A line segment has the same length, an angle has the same angle measure, and a **geometric** figure has the same shape and size as itself.

## What is an example of the reflexive property?

We learned that the **reflexive property** of equality means that anything is equal to itself. … This **property** tells us that any number is equal to itself. For **example**, 3 is equal to 3.

## Why is reflexive property important?

**Reflexive property**in proofs

The **reflexive property** can be used to justify algebraic manipulations of equations. For example, the **reflexive property** helps to justify the multiplication **property** of equality, which allows one to multiply each side of an equation by the same number.

## Reflexive Property Of Equality

**Reflexive Property**

The Reflexive Property states that for every real number xx, x=xx=x .

**Symmetric Property**

The Symmetric Property states that for all real numbers x and yx and y ,

if x=yx=y , then y=xy=x .

**Transitive Property**

The Transitive Property states that for all real numbers x ,y, and z,x ,y, and z,

if x=yx=y and y=zy=z , then x=zx=z .

**Substitution Property**

If x=yx=y , then xx may be replaced by yy in any equation or expression.

## Reflexive Property Definition

A binary relation is called **irreflexive**, or **anti-reflexive**, if it doesn’t relate any element to itself. An example is the “greater than” relation (*x* > *y*) on the real numbers. Not every relation which is not reflexive is irreflexive; it is possible to define relations where some elements are related to themselves but others are not (i.e., neither all nor none are). For example, the binary relation “the product of *x* and *y* is even” is reflexive on the set of even numbers, irreflexive on the set of odd numbers, and neither reflexive nor irreflexive on the set of natural numbers.

A relation ~ on a set *X* is called **quasi-reflexive** if every element that is related to some element is also related to itself, formally: ∀ *x*, *y* ∈ *X* : *x* ~ *y* ⇒ (*x* ~ *x* ∧ *y* ~ *y*). An example is the relation “has the same limit as” on the set of sequences of real numbers: not every sequence has a limit, and thus the relation is not reflexive, but if a sequence has the same limit as some sequence, then it has the same limit as itself. It does make sense to distinguish **left** and **right quasi-reflexivity**, defined by ∀ *x*, *y* ∈ *X*: *x* ~ *y* ⇒ *x* ~ *x*^{} and ∀ *x*, *y* ∈ *X* : *x* ~ *y* ⇒ *y* ~ *y*, respectively. For example, a left Euclidean relation is always left, but not necessarily right, quasi-reflexive.

A relation ~ on a set *X* is called **coreflexive** if for all *x* and *y* in *X* it holds that if *x* ~ *y* then *x* = *y*.^{} An example of a coreflexive relation is the relation on integers in which each odd number is related to itself and there are no other relations. The equality relation is the only example of a both reflexive and coreflexive relation, and any coreflexive relation is a subset of the identity relation. The union of a coreflexive and a transitive relation is always transitive.

## What Is The Reflexive Property

Why is the reflexive property of equality important or even necessary to state? After all, it seems so obvious! The reason is that if we don’t clearly make a statement of something in mathematics, how do we know that we all agree that it is true? Even for something so simple as the reflexive property of equality, we need to have a property so that we know that we all agree that *x* = *x*.

Also, if we did not have the reflexive property of equality, how would we explain that *x* < *x* or *x* > *x* is not true? Because of this property of equality, we can affirm that statements like *x* < *x* are false.

## Reflexive Property Geometry

The reflexive property of equality means that all the real numbers are equal to themselves. This property is applied for almost every number. It is used to prove the congruence in geometric figures. The reflexivity is one of the three properties that define the equivalence relation. Determine what is the reflexive property of equality using the reflexive property of equality definition, for example, tutorial.

The reflexive property of equality simply states that value is equal to itself.

Further, this property states that for all real numbers, x = x.

Real numbers include all the numbers on a number line. They include rational numbers and irrational numbers. A rational number is any number that can be written as a fraction. An irrational number, on the other hand, is a real number that cannot be written as a simple fraction. Square roots would be in this category. In fact, real numbers pretty much entail every number possible except for negative square roots because they are imaginary numbers.

## Reflexive Property In Geometry

First, **Real numbers are an ordered set of numbers**. This means real numbers are sequential. The numerical value of every real number fits between the numerical values of two other real numbers.

Everyone is familiar with this idea since all measurements (weight, the purchasing power of money, the speed of a car, etc.) depend upon the fact that some numbers have a higher value than other numbers. Ten is greater than five, and five is greater than four . . . and so on. This is an important math property.

Second, **we never run out of real numbers**. The quantity of real numbers available is not fixed. There are an infinite number of values available. The availability of numbers expands without end. Real numbers are not simply a finite “row of separate points” on a number line. There is always another real number whose value falls between any two real numbers (this is called the “density” property).

Third, **when real numbers are added or multiplied, the result is always another real number** (this is called the “closure” property). [This is not the case with all arithmetic operations. For example, the square root of -1 yields an imaginary number.]

With these three points in mind, the question is: , How can we use real numbers in practical calculations? What rules apply?

- How should numbers be added, subtracted, multiplied, and divided? What latitude do we have?
*Does it matter what we do first?*second? third?..*Can we add*a series of numbers together*in any order?*Will the final answer be the same regardless of the order we choose?*Can we multiply*a series of numbers together*in any order?*Will the final answer be the same regardless of the order we choose?

*The* following *properties of real numbers answer these types of questions.* The property characteristics which follow show how much latitude you have to change the mechanics of calculations which use real numbers without changing the results.

- Associative Property
- Commutative Property
- Distributive Property
- Identity Property
- Inverse Property

In addition, the following *three math properties of equivalence* determine when one algebraic quantity can be substituted for another without changing the original value.

- Reflexive Property
- Symmetric Property
- Transitive Property