Reflexive Property

In algebra, we study the reflexive property of different forms such as the reflexive property of equality, reflexive property of congruence, and reflexive property of relations. Reflexive property works on a set when every element of the set is related to itself. The reflexive property of equality is applied to the set of numbers which states that every number is equal to itself. On the other hand, the reflexive property of congruence states that any geometric figure is congruent to itself. Let us learn more about the reflexive property with the help of examples for a better understanding.

The reflexive property on a set states that every element of the set is related to itself. If the relation defined on a set is congruence, then it is called the reflexive property of congruence and if the relation defined on a set of numbers is equality, then it is called the reflexive property of equality. In such a case, we can call the relation defined to be reflexive relation or reflexive property is satisfied on that set. Let us now understand the reflexive property of equality and congruence in the following sections.

The reflexive property of congruence in geometry states that every angle, every line, and every figure/shape is congruent to itself. This property is generally used in proofs such as proving two triangles are congruent and in proofs of parallel lines. If two triangles share a common side or a common angle, then we can use the reflexive property of congruence to prove the two triangles are congruent.

Consider the figure given below. Two triangles ABC and CDA are joined together and have a common side AC. To show the two triangles are congruent, we are given AB = AD and BC = CD. Using the reflexive property of congruence, since every line segment is congruent to itself, we have AC = AC. So, the two triangles are congruent by the SSS congruence rule.

The reflexive property of equality states that every number is equal to itself. It is a relation defined on the set of numbers as aRb if and only of a = b, for all numbers a and b. We can write this reflexive property as, if x is a number, then x = x. This property of reflexivity is used to prove equivalence relations defined on the set of numbers.

Consider a relation R defined on the set of real numbers as aRb if and only if a = b. To prove R is an equivalence relation, we will prove that R is reflexive, symmetric, and transitive.

• Using the reflexive property of equality, we know that every number is equal to itself, so a = a which implies aRa for all real a.
• If aRb, then a = b, and we know b = a ⇒ bRa. So, R is symmetric.
• If aRb and bRc, then a = b and b = c which implies a = c. So, aRc. Therefore, R is transitive.

Hence, R is an equivalence relation. This is how the reflexive property of equality is used to prove an equivalence relation.

A binary relation R defined on a set A is said to be reflexive if, for every element a ∈ A, we have aRa, that is, (a, a) ∈ R. In other words, we can say that a relation defined on a set is a reflexive relation if and only if every element of the set is related to itself. Reflexive property of equality and congruence are special cases of reflexive property of relations.

Important Notes on Reflexive Property

• The reflexive property on a set states that every element of the set is related to itself.
• Reflexive property is of different forms such as the reflexive property of equality, reflexive property of congruence, and reflexive property of relations.
• A binary relation R defined on a set A is said to be reflexive if, for every element a ∈ A, we have aRa, that is, (a, a) ∈ R.

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What is Reflexive Property in Algebra?

In algebra, we study the reflexive property of different forms such as the reflexive property of equality, reflexive property of congruence, and reflexive property of relations. Reflexive property works on a set when every element of the set is related to itself.

What is Reflexive Property in Geometry?

in geometry, the reflexive property states that any geometric figure is congruent to itself. Every angle, every line, and every figure/shape is congruent to itself.

What is Reflexive Property in Triangle?

The reflexive property in triangles basically implies that each angle and side of a triangle is equal to itself. Hence, we have that every triangle is congruent to itself.

How Do You Use Reflexive Property?

We can use the reflexive property of equality to show that 'is equal to' on a set of numbers is an equivalence relation. We use the reflexive property of congruence to show two triangles are congruent.

What is Reflexive Property of Equality?

The reflexive property of equality states that every number is equal to itself. It is a relation defined on the set of numbers as aRb if and only of a = b, for all numbers a and b.

What is Reflexive Property of Congruence?

The reflexive property of congruence in geometry states that every angle, every line, and every figure/shape is congruent to itself. This property is generally used in proofs such as proving two triangles are congruent.

When Do We Use the Reflexive Property of Relations?

A binary relation R defined on a set A is said to be reflexive if, for every element a ∈ A, we have aRa, that is, (a, a) ∈ R. We use this to show if a relation is reflexive or equivalence.