Identity relation vs Reflexive Relation

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 $R$.

Solution 1:

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:

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

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.

Solution 2:

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.