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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 $R$.

M47145
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Rickybobby
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    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

3 Answers3

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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
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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
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    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
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    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
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    @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
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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
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