Show that the relation $$R$$ in the set $$A$$ of all the books in a library of a college, given by ( $R = \{ (x, y) : x\;$ and $\; y\;$ have same number of pages$\}$ ) is an equivalence relation.

Toolbox:
• A relation R in a set A in an equivalance relation. if R is reflexive, symmetric and transitive.
Let $A=$Set of all book in library in college. Given $R=$ {$(x,y):x$ and $y$ have same number of pages}:
If $x=y$, such that $(x,x) \in R$, then $x$ and $y$ have the same number of pages. Hence, $R$ is reflexive.
For $R$ to be symmetric, $(x,y) \in R \; \Rightarrow \; (y,x) \in R$
If $x$ has same number of pages as $y$ then $y$ also has same number pages as $x$. Therefore, $R$ is symmetric
For $R$ to be transitive, if $(x,y),(y,z) \in R \; \Rightarrow \; (x,z) \in R$
If x and y have same number of pages and y and z have same number of pages Then x and z also have same number of pages. Hence R is transitive
Hence R is an equivalance relation.
answered Feb 21, 2013 by
edited Mar 8, 2013
{"extraction_info": {"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": 1, "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, "math_score": 0.8249318599700928, "perplexity": 272.04284142605707}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886105970.61/warc/CC-MAIN-20170820034343-20170820054343-00349.warc.gz"}