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# Reflexive Property Of Equality

The reflexive property of equality is a fundamental property of mathematics that states that any quantity or expression is equal to itself. In other words, for any real numbers, variables, or expressions a, a = a.

This property is often used in mathematical proofs and serves as the basis for many other properties of equality. For example, the symmetric property of equality states that if a = b, then b = a, and the transitive property of equality states that if a = b and b = c, then a = c.

The reflexive property is also applicable in other areas of mathematics, such as set theory, where it is used to define the identity element of a set. In set theory, the identity element is an element that, when combined with another element of the set using a particular operation, leaves the other element unchanged.

Overall, the reflexive property of equality is a simple yet essential concept in mathematics that forms the basis for many other properties and helps to establish the fundamental rules of mathematical reasoning.

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.

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.

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.

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 xandyx  and  y ,

if x=yx=y , then y=xy=x .

Transitive Property

The Transitive Property states that for all real numbers x,y,andz,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: ∀ xy ∈ 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 ∀ xy ∈ X: x ~ y ⇒ x ~ x and ∀ xy ∈ 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.

In this lesson, we will review the definition of the reflexive property of equality. We will also look at why this property is important. Following the lesson will be a brief quiz to test your knowledge on the reflexive property of equality.

## 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

FirstReal 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.

Secondwe 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).

Thirdwhen 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