Reflexive Property

1. Defining the Reflexive Property of Equality

In the algebra, the reflexive property of the equality states that a specific number can always said to be equal to itself. Therefore, if a would be a number, then. a = a.

2. Importance of the Reflexive Property of Equality

A. Reflexive Property

The reflexive property of the equality states that a specific number can always said to be equal to itself. Therefore, if a would be a number, then. a = a

B. Symmetric Property

The Symmetric Property actually states that for every real number x and y , if x = y , then y = x.

C. Transitive Property

In the Transitive Property, the Transitive Property actually mentions that in case of every real number x, y, and z, if x = y and y = z, then x = z.

D. Substitution Property

The substitution property of the equality, 1 of the 8 properties of the equality, actually mentions that if x = y, then it shall be possible to substitute x for y in any specific equation, and it shall be possible to substitute y for x in any specific equation.

3. Examples of Reflexive Property of Equality

Reflexive Property

The above-said property might seem certain and obvious, although, it can be considered to be very important. It can be considered to be one of the fundamental truths that actually makes every math work.

The reflexive property in relation to equality actually says or mentions that a number is equal to itself, for instance, a = a.

It should be noted that the reflexive property is just like looking in a specific mirror. Whatever one places in front shall be shown or demonstrated in the reflection.

This can be said to be very true in relation to any number, whether real or imaginary.

2=24i=4i

This can also say to be true regarding the variables, provided that it is the same specific variable that is written in the same or similar equation.

x=x15y=15y10r2=10r2

The reflexive property shall be possible to utilize in order to justify the algebraic manipulations regarding the equations. For instance, the reflexive property actually helps or enables to justify the particular multiplication property of the equality, which actually enables a person to multiply every side of any specific equation by the same or similar number.

Let a,a, and bb be numbers like that a=b.a=b. Prove that if cc is actually a number, then ac=bc.ac=bc.

Statements	Reasons
1. a=ba=b	1. Given
2. ac=acac=ac	1. Reflexive property of the equality
3. ac=bcac=bc	3. Substitution property of the equality

The reflexive property of congruence is normally utilized in the geometric proofs when the certain congruences are actually required to be established. For instance, in order to prove that 2 triangles are congruent, 3 congruences shall be required to be established (SSS, SAS, ASA, AAS, or HL properties of the congruence). If any side is shared amidst the triangles, then the specific reflexive property shall be required to demonstrate the congruence of the side with itself.

Given that overline{AB} cong overline{AD}AB≅AD and overline{BC} cong overline{CD},BC≅CD, prove that triangle ABC cong triangle ADC.△ABC≅△ADC.

 

Statements	Reasons
1. overline{AB} cong overline{AD};AB≅AD; overline{BC} cong overline{CD}BC≅CD	1. Given
2. overline{AC} cong overline{AC}AC≅AC	1. Reflexive property of the congruence
3. triangle ABC cong triangle ADC△ABC≅△ADC	3. The SSS triangle congruence

4. Using the Reflexive Property to Prove Other Properties of Equality

The Reflexive Property of the Equality actually means that any quantity can be said to be equal to itself as well as in the same order. It can be said to be very useful as it delivers the clarity as well as consistency in the math for the individuals to know that a certain thing is equal in relation to itself (4=4, x=x, a+b=a+b).

This specific property is actually utilized primarily in geometry in order to help in the proofs by mentioning that a certain thing is actually equal to itself in order to help prove any other statement in any geometric proof.

Reflexive Property of the Equality Examples 

It should be noted that the equals sign with tilde on the top actually means that the 2 quantities are actually congruent.

For instance, a proof might actually state that line segment AC is actually equal to the line segment AC, which might be useful in proving or demonstrating the congruence of the 2 connected shapes.

Example of the Reflexive Property of the Equality with Triangles

Suppose there is more knowledge regarding the other parts or portions in relation to the triangles (the congruent angles or the sides). In that case, the triangles might be proven congruent with the help of the utilization of the Reflexive Property of the Equality with the AC.

5. Difference between Reflexive and Symmetric in Discrete Math

The reflexive property of equality states that a specific number can always said to be equal to itself. Therefore, if a would be a number, then. a = a.

One should certainly understand the following: -

Reflexive -> <a,a=a>, <b,b=b> uses = to describe.

Symmetric -> <a,b>, <b,a> uses ≤, ≥, = to describe.

In this regard, it should be noted that a relation shall be considered to be reflexive if, for every aa, aa can be said to be ‘related’ to aa (aRaaRa is actually true).

A relation is actually symmetric if, whensoever aRb,aRb, then the bRabRa (so ≤≤ is not symmetric the reason being that if a≤ba≤b then it shall not follow that b≤ab≤a, but equality shall be symmetric).

Symmetric actually means that in relation to every (a,b)∈R(a,b)∈R also (b,a)∈R(b,a)∈R

(c,c)(c,c) is actually symetric with itself.

In this regard, it can be said that reflexive actually means that such (c,c)(c,c) actually exists at all, as well as does not necessarily shall mean that the specific relation is actually symmetric.