19

So we're starting relations in my discrete structures class this week, and I've probably read this over 10 times by now...I believe I have a good understanding of Identity Relations, but Reflexive Relations seem to have me slightly confused.

From my understanding, an example of Identity relation using set $A = \{1,2,3,4\}$

  1. $R_1 = \{(1,1), (2,2), (3,3), (4,4)\}$ because each element is equal to itself.
  2. $R_2 =\{ (1,1), (2,2), (3,3), (4,4), (1, 4)\}$ would not be an identity relation, as $1 \neq 4$.

What I don't understand is why

The relation $R_2$ defined by $R_2 = \{(1, 1), (3, 3), (2, 1), (3, 2)\}$ is not a reflexive relation on $A$, since $(2, 2) \notin R_2$.

is not a Reflexive Relation

Could someone give me an example of what a simple reflexive relation is, and isn't?

Thanks all for the input, see below for a good example of a Reflexive Relation

Here's what the book describes both as:

Identity relation.

Let $A$ be any set. Then the relation $R = \{(x, x) : x \in A\}$ on $A$ is called the identity relation on $A$. Thus, in an identity relation, every element is related to itself only.

For example, consider $A = \{a, b, c\}$ and define relations $R_1$ and $R_2$ as follows. $R_1 = \{(a, a) ,(b, b), (c, c)\}$
$R_2 = \{(a, a), (b, b), (c, c), (a, c)\}$

Then $R_1$ is an identity relation on $A$, but $R_2$ is not an identity relation on $A$ as the element $a$ is related to $a$ and $c$.

Reflexive relation.

A relation $R$ on a set $A$ is said to be a reflexive relation if every element of $A$ is related to itself. Thus, $R$ is reflexive iff $(x, x) \in R$ for all $x \in A$. A relation $R$ on a set $A$ is not reflexive if there is an element $x \in A$ such that $(x, x) \notin R$. For example, consider $A = (1, 2, 3)$. Then the relation $R_1$ defined by $R_1 = \{(1, 1), (2, 2), (3, 3), (1, 3), (2, 1)\}$ is a reflexive relation on $A$. The relation $R_2$ defined by $R_2 = \{(1, 1), (3, 3), (2, 1), (3, 2)\}$ is not a reflexive relation on $A$, since ($2, 2) \notin R_2$. Remark

Every identity relation on a non-empty set $A$ is a reflexive relation, but not conversely. Consider $A = \{a, b, c\}$ and define a relation $R$ by $R = \{(a, a), (b, b), (c, c), (a, b)\}$. Then $R$ is a reflexive relation on $A$ but not an identity relation on $A$ due to the element $(a, b)$ in $R$.

RobPratt
  • 50,938
Rickybobby
  • 377
  • 5
    If you keep asking questions like this, with this level of detail and patience, you will be a fabulous mathematician. – The Count Jun 23 '16 at 00:36
  • I recommend reading my answer here where I give a graph theoretic way to interpret the definitions of reflexivity, symmetry, antisymmetry, and transitivity. – JMoravitz Jun 23 '16 at 00:40
  • An identity relation is just a special case of a reflexive relation that contains no further data. That is, you can think of the identity relation on a set as the "smallest" reflexive relation on the given set. We can formalize this intuition in many ways, one of them is to say that an identity relation on a set $A$ is the intersection of all reflexive relations on $A$. – Bruno Bentzen Jun 21 '17 at 07:37

4 Answers4

17

A relation $R$ on $A$ is reflexive if $(x,x)\in R$ for every $x\in A$.

So if $A=\{1,2,3,4\}$ the following are all reflexive:

  1. $R=\{(1,1), (2,2), (3,3), (4,4)\}$
  2. $R=\{(1,1), (1,2), (2,2), (3,3), (4,4)\}$
  3. $R=\{(1,1), (1,3), (2,2), (2,3), (2,4), (3,3), (4,1), (4,4)\}$

Each of the above contains $(1,1),(2,2),(3,3)$ and $(4,4)$, making them reflexive. Note that 1. is the identity, and it is reflexive. However, the following are not reflexive:

  1. $R=\{(1,1), (2,2), (3,1), (4,4)\}$
  2. $R=\{(1,1)\}$
  3. $R=\{(1,1), (1,3), (1,4),(2,1), (2,2), (3,1), (3,3), (4,3)\}$

In 4. $R$ does not contain $(3,3)$ so it is not reflexive. In 5. $R$ does not contain $(2,2), (3,3),$ or $(4,4)$ so it is not reflexive. In 6. $R$ does not contain $(4,4)$, and hence it not reflexive either.

M47145
  • 4,226
6

Re-read your definition of a reflexive relation $R$: Every element must be related (under $R$) to itself. In your example, since we don't have $R(2,2)$ $R$ can't meet this definition.

Mark Fischler
  • 42,297
  • 1
    So if I'm understanding correctly... with my previous example of identity relation of A = {1,2,3,4) would be

    R1 = (1,1) (2,2) (3,3) (4,4) is an identity relation R2 = (1,1) (2,2) (3,3) (4,4) (1,4) is not a identity relation, but reflexive?

     – Rickybobby Jun 23 '16 at 00:40
  • @beatles1235 your R2 is indeed reflexive. You will find that it is also transitive and antisymmetric, but not symmetric. In fact, all reflexive relations contain the identity relation as a subset. – JMoravitz Jun 23 '16 at 00:45
  • 1
    So with reflective relation, I can have an element that is not related to itself, as long as i have at least all elements related to itself. Does that sound correct? – Rickybobby Jun 23 '16 at 00:47
  • @beatles1235 Your example has all elements related to itself. It also has an element related to a different nonequal element. Just because one of the comparisons (in this case (1,4)) is between two unequal things, the fact that all are related to themselves does not change. A better way to say your first line is "I can have an element that is related to an element other than itself" – JMoravitz Jun 23 '16 at 00:49
  • perfect! if you feel like putting that into an answer, i'd totally accept that!! thank you for helping me understand that! – Rickybobby Jun 23 '16 at 00:50
  • 1
    @beatles1235: You may also find the following observation helpful. If $D$ is the identity relation on a set $A$, then a relation $R$ on $A$ is reflexive if and only if $D\subseteq R$. That $\subseteq$ means that $R$ has to contain all of the pairs $\langle a,a\rangle$ with $a\in A$, but it can contain other pairs as well. – Brian M. Scott Jun 23 '16 at 00:52
0

Again to clerify, in few words

An identity relation is always reflexive, but a reflexive relation is not always identity relation.

the key is in the definition, it clearly states, that in identity relation element is only related to itself, but in relexive no such thing is said, it must be related to itself, but it can also be related to others.

Example - for a relation on set $A = \{1,2,3,4\}$

$R_1 = \{(1,1),(1,2),(2,2), (3,3),(4,4), (4,3)\}$ is reflexive, but not identity

but

$R_2 = \{(1,1),(2,2), (3,3),(4,4)\}$ is reflaxive and identity.

Yanjan. Kaf.
  • 310
-2

Let, a,b € Z (a, a) € R. Then, a-a=0 =(a-a) is divisible by 2 Z={1,2,3}. So, Z×Z =R={(1,1),(2,2),(3,3),(1,3). ["R" is reflexive relation] I={(1,1),(2,2),(3,3)}. [where, "I" is Identity Relation] So,from the above example we can notice that :- Reflexive relation- is a kind of relation which contains the elements related to itself as well as can contain other pairs too. Identity Relation- is a kind of relation which contains the elements related to itself only. **Thus, "Every IDENTITY Relation on a Non-Empty set is a REFLEXIVE Relation but not vice-versa....

Dr. Swapnanil Maity
  • 1