# Is a relation that is purely reflexive also symmetric?

Mathematics Asked by Paul J on September 13, 2020

Is a relation that is pruely reflexive also symmetric?

For example, say you have a relation defined as $$R = {(a,a),(b,b)}$$. This is purely reflexive, but is it also symmetric? The typical symmetric definition is $$aRb Leftrightarrow bRa$$, which is kinda shown in this as $$aRa Leftrightarrow aRa$$, but I am unsure. Sorry if this is a trivial question, I am just learning about this stuff in a proof course and am slightly confused.

I've seen a useful method for getting a more intuitive understanding of some of these properties of relations, namely using a table to indicate a relation and then visually deduce the property. In the case you describe one obtains:

For example, if a relation is reflexive, the diagonal elements will all be populated with 1's. Note that a "1" indicates that the corresponding row entry and column entry is in the relation, and a "0" means that combination in not in the relation. For example, $$left(a,aright) in R$$, but $$left(a,bright) notin R$$, etc. in the above table.

If the relation is symmetric, then the table will identical if you reflect it about the diagonal. Therefore, for the relation you have described in the question, this is clearly true.

Here is another question in Stack Exchange that illustrates this idea further: Checking the binary relations, symmetric, antisymmetric and etc

I hope this helps.

## Related Questions

### why infinite carrying does not occur

2  Asked on December 12, 2020 by ken-wong

### Is this a reasonable measure for comparing the sizes of two sets of real numbers?

1  Asked on December 11, 2020 by samuel-muldoon

### Under which condition will a Euler graph’s complement also be a Euler graph?

0  Asked on December 11, 2020 by isaac_42

### Summations with Fractional Indices (Future Value)

1  Asked on December 11, 2020 by jippyjoe4

### How are the Taylor series taken?

0  Asked on December 10, 2020

### Show that $Vert vVert^2= langle v,AA’ vrangle iff AA’=I$

1  Asked on December 10, 2020 by chuck

### How can you approach $int_0^{pi/2} xfrac{ln(cos x)}{sin x}dx$

3  Asked on December 10, 2020

### Does $limlimits_{xto3} (4 – x)^{tan (frac {pi x} {2})}$ converge?

3  Asked on December 10, 2020 by ethan-mark

### Is Theory of ODEs by Coddington and Levinson still a good source for learning ODEs?

1  Asked on December 10, 2020 by mert-batu

### Find the determinant of a matrix $A$, such that $A^4 + 2A = 0$

2  Asked on December 10, 2020 by avivgood2

### Question about the correspondece between unitary Mobius transformations and quaternions.

0  Asked on December 10, 2020 by user2554

### How do we show $P(A) leq P(A Delta B) + P(A cap B) leq P(ADelta B) + P(B).$?

1  Asked on December 10, 2020 by user825841

### Prove that $H<GRightarrow |H|le leftlfloor frac{|G|}{2}rightrfloor$ without Lagrange's theorem.

1  Asked on December 10, 2020

### Cauchy-Schwartz Inequality problem

2  Asked on December 10, 2020 by in-finite

### Solving the Diophantine equation $x^2 +2ax – y^2 = b$ in relation to Integer Factoring

0  Asked on December 10, 2020 by vvg

### Subgroup element composition

3  Asked on December 10, 2020 by conchild

### Motivation of Jacobi symbol

0  Asked on December 9, 2020 by athos

### Convergence of points under the Mandelbrot iterated equation

1  Asked on December 9, 2020 by robert-kaman

### Finding the quadrilateral with a fixed side and perimeter that has a maximum area

1  Asked on December 9, 2020 by ak2399

### Convergence of series $sum_{n=1}^inftyleft[left(1+frac{1}{n}right)^{n+1}-aright]sin{n}$

1  Asked on December 9, 2020 by alans