# Cartesian product

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

The Cartesian product of two sets $A$ and $B$ is the set of all ordered pairs $(a,b)$ such that $a$ is an element of $A$ and $b$ is an element of $B$. More generally, the Cartesian product of an ordered family of sets $A_1, A_2, \dotsc$ is the set $A_1 \times A_2 \times \dotsb$ of ordered tuples $(a_1, a_2, \dotsb)$ such that $a_j$ is an element of $A_j$, for any positive integer $j$ for which we have specified a set $A_j$.
{"ft_lang_label":"__label__en","ft_lang_prob":0.94571733,"math_prob":0.9999496,"size":652,"snap":"2023-40-2023-50","text_gpt3_token_len":167,"char_repetition_ratio":0.14351852,"word_repetition_ratio":0.0877193,"special_character_ratio":0.2714724,"punctuation_ratio":0.0942029,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9988925,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,9,null,9,null,9,null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-12-09T01:47:50Z\",\"WARC-Record-ID\":\"<urn:uuid:065618df-e53f-44e0-8287-ef9340898f66>\",\"Content-Length\":\"37759\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:909aa543-3873-4ed3-8d6a-954ed203aa55>\",\"WARC-Concurrent-To\":\"<urn:uuid:6536079f-8c75-4961-9915-ac377bb29c98>\",\"WARC-IP-Address\":\"104.26.11.229\",\"WARC-Target-URI\":\"https://artofproblemsolving.com/wiki/index.php?title=Cartesian_product&oldid=20200\",\"WARC-Payload-Digest\":\"sha1:BZZHNMFPF2KHSZZEHL3LG3DBJJKP5EV4\",\"WARC-Block-Digest\":\"sha1:UEUYVYWHZ6G3SKCUKK35HPKIASDBSPFE\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679100781.60_warc_CC-MAIN-20231209004202-20231209034202-00389.warc.gz\"}"}