Sets and Relations

[salik]

Hi all, I am having some trouble understanding sets and relations(?) in my Discrete Maths course where it talks about Reflexive, Symmetric and Transitive.. And I am unable to find any good videos (please do tell me should any chance upon a good tutorial)
Please do correct me on the following if I am getting it wrong, as I am writing them in my own terms..

1. Suppose if I have a relation R = {(1,1), (2,2), (3,3)}, this will be called Reflexive..
Basically, the numbers should be relating (same) to each other

2. Suppose if I have a relation R = {(1,2), (2,1), (3,2), (2,3)}, this will be called Symmetric..
If (1,3) is added into R, it will not be Symmetric because (3,1) is missing, am I correct?

3. Suppose if I have a relation R = {(1,2), (2,3), (1,3)}, this will be called Transitive..
I assume that each number can only be related once to other number?

Do anyone has a good example usually how such question will be asked in exam? To be honest, I feel that the above example that I give is pretty weak and may not be practical..

 

[Limax]

1. Suppose if I have a relation R = {(1,1), (2,2), (3,3)}, this will be called Reflexive..
Basically, the numbers should be relating (same) to each other

Be careful. A relation R is generally defined on a set S, and R is reflexive iff (a,a) is in R for every element a in S. In this example, if S = {1,2,3}, the relation is reflexive, but if S = {1,2,3,4} then it is not, as (4,4) is not in it.

2. Suppose if I have a relation R = {(1,2), (2,1), (3,2), (2,3)}, this will be called Symmetric..
If (1,3) is added into R, it will not be Symmetric because (3,1) is missing, am I correct?

Correct.

3. Suppose if I have a relation R = {(1,2), (2,3), (1,3)}, this will be called Transitive..
I assume that each number can only be related once to other number?

It means then whenever (a,b) and (b,c) are in R, then (a,c) must be in R.

PS: It is more usual to write aRb to mean that (a,b) is in R.

 

[pka]

Hi all, I am having some trouble understanding sets and relations(?) in my Discrete Maths course where it talks about Reflexive, Symmetric and Transitive..
1. Suppose if I have a relation R = {(1,1), (2,2), (3,3)}, this will be called Reflexive..
Basically, the numbers should be relating (same) to each other

2. Suppose if I have a relation R = {(1,2), (2,1), (3,2), (2,3)}, this will be called Symmetric..
If (1,3) is added into R, it will not be Symmetric because (3,1) is missing, am I correct?

3. Suppose if I have a relation R = {(1,2), (2,3), (1,3)}, this will be called Transitive..
I assume that each number can only be related once to other number?

Please, please note that relations are defined as a set of ordered pairs. See reply #2.
So we must begin with a set, say \(\displaystyle S\). Next form the cross product, \(\displaystyle S\times S=\{(x,y) : \{x,y\}\subset S\}\).
Now we need the set of all subsets of \(\displaystyle \mathcal{ P}(S\times S)\)

DEFINITION: If \(\displaystyle R\subset \mathcal{ P}(S\times S)\) then \(\displaystyle R \) is a relation in \(\displaystyle S\).

Above I indicated in red some problematic language as I see it.
The diagonal is \(\displaystyle \Delta_R=\{(x,x) : x\in S\}\), because that is a relation call the identity relation on \(\displaystyle S\)( i.e. every element of \(\displaystyle S\) is related to itself).
A relation \(\displaystyle \large R \) is reflexive if an only if \(\displaystyle {\large\Delta_R\subset R} \)

A relation \(\displaystyle \large R \) is symmetric if an only if \(\displaystyle {\large R= R^{-1}} \)

A relation \(\displaystyle \large R \) is transitive if an only if \(\displaystyle (x,y) \in R\,\& \,(y,z) \in R \Rightarrow (x,z) \in R\)

 

[Ishuda]

Hi all, I am having some trouble understanding sets and relations(?) in my Discrete Maths course where it talks about Reflexive, Symmetric and Transitive.. And I am unable to find any good videos (please do tell me should any chance upon a good tutorial)
Please do correct me on the following if I am getting it wrong, as I am writing them in my own terms..

1. Suppose if I have a relation R = {(1,1), (2,2), (3,3)}, this will be called Reflexive..
Basically, the numbers should be relating (same) to each other

2. Suppose if I have a relation R = {(1,2), (2,1), (3,2), (2,3)}, this will be called Symmetric..
If (1,3) is added into R, it will not be Symmetric because (3,1) is missing, am I correct?

3. Suppose if I have a relation R = {(1,2), (2,3), (1,3)}, this will be called Transitive..
I assume that each number can only be related once to other number?

Do anyone has a good example usually how such question will be asked in exam? To be honest, I feel that the above example that I give is pretty weak and may not be practical..

You might check out
https://www.google.com/search?q=reflexive+symetric&ie=utf-8&oe=utf-8#q=reflexive+symmetric&tbm=vid