1. ## Properties of Relations

Hello,

I just have a question concerning properties of relations.
I'm wondering, if we wanna check wheter a relation is symmetric or not, is it ok to consider identical pairs..??

You know like.. R = {(a,b) | a^2 = b^2}.. R is defined on the integers.
Now the way I'm thinking of this is like.. lets take a look at (1,2) & (2,1) obviously the square of 1 doesnt equal the square of 2 so its not symmetric. But can we instead, consider a pair like (1,1) or (2,2).. which in that case can make the relation symmetric...???

Another question... can a relation be not symmetric and not anti-symmetric at the same time ???

I hope somebody can clarify theses point.

2. Let $\displaystyle R$ be a binary relation.
$\displaystyle R$ is symmetric if $\displaystyle aRb\Rightarrow bRa, \ \forall a,b$.

If $\displaystyle aRb\Rightarrow a^2=b^2\Rightarrow b^2=a^2\Rightarrow bRa$
So, the relation is symmetric.

3. Originally Posted by MMM88
Hello,

I just have a question concerning properties of relations.
I'm wondering, if we wanna check wheter a relation is symmetric or not, is it ok to consider identical pairs..??

You know like.. R = {(a,b) | a^2 = b^2}.. R is defined on the integers.
Now the way I'm thinking of this is like.. lets take a look at (1,2) & (2,1) obviously the square of 1 doesnt equal the square of 2 so its not symmetric. But can we instead, consider a pair like (1,1) or (2,2).. which in that case can make the relation symmetric...???

You relation $\displaystyle R$ is symmetric because if $\displaystyle (a,b) \in R$, then so is $\displaystyle (b,a)$

A specific pair can show that a relation is not symmetric, but cannot show
that it is symmetric.

Also a relation is not defined on all pairs. In this case $\displaystyle (a,b) \in R$ if and only
if $\displaystyle a^2=b^2$, $\displaystyle R$ has nothing to say about any other pairs.

RonL

4. Originally Posted by MMM88

Another question... can a relation be not symmetric and not anti-symmetric at the same time ???
$\displaystyle R=((1,2),(2,1),(3,4))$

is not anti-symmetric because $\displaystyle (1,2) \in R$ as is $\displaystyle (2,1)$, also it is not symmetric
because $\displaystyle (3,4) \in R$ but $\displaystyle (4,3) \not\in R$

RonL

5. Awesome!!! I get it now.
Thanks for the help guys.
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "http://mathhelpforum.com/discrete-math/23871-properties-relations.html", "fetch_time": 1527473147000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2018-22/segments/1526794870604.65/warc/CC-MAIN-20180528004814-20180528024814-00056.warc.gz", "warc_record_offset": 178629268, "warc_record_length": 10401, "token_count": 638, "char_count": 2307, "score": 4.1875, "int_score": 4, "crawl": "CC-MAIN-2018-22", "snapshot_type": "latest", "language": "en", "language_score": 0.8782187104225159, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00022-of-00064.parquet"}