What Is A Symmetric Property

Symmetric Property

The symmetric property is an essential property in algebra that is used in diverse math concepts such as equality, matrices, relations, congruence, etc. In full general, the symmetric property on a fix states that if one chemical element is related to the second chemical element, the 2d element is also related to the first chemical element co-ordinate to the relation divers on the gear up.

In this article, we will draw the symmetric property of equality, symmetric property of congruence, symmetric property of relations, and the symmetric holding of matrices. We will solve various examples related to the symmetric property to better understand the concept.

1.	What is Symmetric Property?
2.	Symmetric Property of Equality
3.	Symmetric Property of Congruence
4.	Symmetric Property of Matrix
5.	Symmetric Property of Relations
6.	FAQs on Symmetric Property

What is Symmetric Holding?

The symmetric property in algebra is defined as a holding that implies if one chemical element in a set up is related to the other, then we can say that the second element is also related to the first element. Nosotros study dissimilar forms of symmetric properties as given beneath:

Symmetric Property of Equality
Symmetric Holding of Congruence
Symmetric Property of Relations
Symmetric Holding of Matrices

symmetric property

All symmetric properties are specific cases of the symmetric holding of relations. For example, if nosotros define a relation on the set of numbers as 'is equal to', then nosotros get the symmetric property of equality. If the relation is divers on the prepare of geometric figures/triangles, then nosotros get the symmetric property of congruence. Let us discuss the symmetric property of equality in the next department.

Symmetric Property of Equality

Now that we have understood the basic meaning of the symmetric property, permit us describe the symmetric property of equality. It states that if a existent number ten is equal to a real number y, then we tin can say that y is equal to ten. This property helps in proving the relation defined on the set of numbers as 'aRb if and just if a = b' to be symmetric relation. Mathematically, we can express the symmetric property of equality every bit 'If 10 = y, then y = x'. This property also helps in finding the value of variables in a system of equations.

Symmetric Holding of Congruence

The symmetric property of congruence states that if a geometric figure is congruent to another, and so we can say that the second figure is congruent to the first figure. For instance, if triangle ABC is congruent to the triangle PQR, and so we can also say that triangle PQR is coinciding to the triangle ABC. Some other case of the symmetric belongings of congruence is that if a line segment AB is congruent to another line segment CD, then we can say that CD is congruent to AB. We can use this property to angles, and other geometric figures.

Symmetric Property of Matrix

The symmetric property of a matrix states that if matrix A is symmetric, then matrix A is equal to its transpose, that is, A = A^T. Some of the important properties of a symmetric matrix are:

Symmetric matrices are commutative, that is, if A and B are symmetric matrices, then AB = BA.
The sum and departure of 2 symmetric matrices give the resultant a symmetric matrix.
For integer due north, if A is symmetric, ⇒ A^north is symmetric.
If a matrix A is symmetric, and so A^-1 is symmetric.

Symmetric Property of Relations

A relation R divers on a fix A is said to be symmetric if for all a, b in A, if aRb then we must have bRa, that is, if (a, b) is in R, so (b, a) is in R. This symmetric property of relations is used to bear witness if a relation R is symmetric or an equivalence relation. The number of symmetric relations for a set having 'n' number of elements is given as N = two^{n(northward+i)/ii}, where Northward is the number of symmetric relations and due north is the number of elements in the ready.

Important Notes on Symmetric Property

Symmetric holding of equality, symmetric belongings of congruence, symmetric belongings of relations, and the symmetric property of matrices are the unlike symmetric properties that we report in algebra.
The symmetric belongings of equality states that if a real number x is equal to a real number y, then we can say that y is equal to x.

☛ Related Articles:

Transitive Property
Transitive Holding of Congruence
Improver Property of Equality

Symmetric Holding Examples

Example 1: If 3x + y = four, and so what is 4 equal to?

Solution: Using the symmetric belongings of equality, nosotros have 3x + y = 4 ⇒ 4 = 3x + y

Answer: 4 is equal to 3x + y.
Example 2: If the line segment AB is congruent to the line segment CD, and AB = v cm, so find the length of CD.

Solution: Since the line segment AB is congruent to the line segment CD, therefore we take AB = CD and AB = 5 cm.

Using the symmetric property of congruence, nosotros tin can write CD = AB = 5cm. So, CD = 5cm

Answer: The length of CD is v cm.
Example 3: If Mary'due south height is equal to that of Jane's, can nosotros say that Jane's height is equal to that of Mary'south?

Solution: Since Mary's height is equal to that of Jane'south, nosotros can write Mary = Jane. Now, using the symmetric property of equality, we can write Jane = Mary.

Respond: Yeah, we can say that Jane'due south meridian is equal to that of Mary's.

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Symmetric Property Questions

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FAQs on Symmetric Belongings

What is Symmetric Holding in Geometry?

The symmetric property in geometry states that if one figure is congruent to some other, so we can say that the second figure is congruent to the first figure. For instance, if angle A is congruent to bending B, then we can say that bending B is congruent to angle B.

What is Symmetric Property of Equality?

The symmetric property of equality states that if a real number 10 is equal to a real number y, and so nosotros can say that y is equal to ten. Mathematically, we tin express the symmetric belongings of equality as 'If 10 = y, then y = x'.

What is Symmetric Property of Congruence?

The symmetric holding of congruence states that if a geometric figure is coinciding to some other, then nosotros can say that the second figure is congruent to the commencement figure.

How Do You Prove Symmetric Belongings?

We tin prove the symmetric property of relation by assuming an element is related to another and and then testify that the latter is likewise related to the quondam.

What is Symmetric Property of an Bending?

The symmetric property of an angle states that if bending A is congruent to bending B, then we tin say that angle B is also congruent to bending A.

What is the Deviation Between the Reflexive Property and Symmetric Property?

The reflexive property states that every element is related to itself and the symmetric belongings states that if one element is related to the 2nd element, the 2d element is likewise related to the first element according to the relation defined on the fix.

What is an Example of Symmetric Property?

Some of the of import examples of symmetric properties are:

If 10 + y = 3, then 3 = x + y
If triangle ABC is coinciding to triangle XYZ, then triangle XYZ is coinciding to triangle ABC.
If matrix A is symmetric, then information technology is equal to its transpose.