{"id":4042,"date":"2021-08-15T11:36:15","date_gmt":"2021-08-15T11:36:15","guid":{"rendered":"https:\/\/mathemerize.com\/?p=4042"},"modified":"2022-01-16T16:56:46","modified_gmt":"2022-01-16T11:26:46","slug":"cartesian-product-of-sets","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/cartesian-product-of-sets\/","title":{"rendered":"What is Cartesian Product of Sets – Definition and Example"},"content":{"rendered":"

Here you will learn what is cartesian product of sets and what is relation and inverse relation with example.<\/p>\n

Let’s begin –<\/p>\n

Cartesian Product of Sets<\/h2>\n

The cartesian product of two sets A, B is a non-void set of all ordered pair (a,b),<\/p>\n

where a \\(\\in\\) A and b \\(\\in\\) B. This is denoted by A \\(\\times\\) B.<\/p>\n

\n

\\(\\therefore\\)   A \\(\\times\\) B = {(a,b) \\(\\forall\\) a \\(\\in\\) A and b \\(\\in\\) B}<\/p>\n<\/blockquote>\n

e.g.   A = {1,2}, B = {a,b}<\/p>\n

A \\(\\times\\) B = {(1,a), (1,b), (2,a), (2,b)}<\/p>\n

Note :<\/strong><\/p>\n

(i)  A \\(\\times\\) B \\(\\ne\\) B \\(\\times\\) A    (Non-commutative)<\/p>\n

(ii) n(A \\(\\times\\) B) = n(A)n(B) and n(P(A \\(\\times\\) B)) = \\(2^{n(A)n(B)}\\)<\/p>\n

(iii) A = \\(\\phi\\) and B = \\(\\phi\\) \\(\\iff\\) A \\(\\times\\) B = \\(\\phi\\)<\/p>\n

(iv) If A and B are two non-empty sets having n elements in common, then (A \\(\\times\\) B) and (B \\(\\times\\) A) have \\(n^2\\) elements in common<\/p>\n

(v) A \\(\\times\\) (B \\(\\cup\\) C) = (A \\(\\times\\) B) \\(\\cup\\) (A \\(\\times\\) C)<\/p>\n

(vi) A \\(\\times\\) (B \\(\\cap\\) C) = (A \\(\\times\\) B) \\(\\cap\\) (A \\(\\times\\) C)<\/p>\n

(vii) A \\(\\times\\) (B – C) = (A \\(\\times\\) B) – (A \\(\\times\\) C)<\/p>\n

Relation<\/h2>\n

Every non-zero subset of A \\(\\times\\) B defined a relation from set A to set B.<br>If R is relation from A \\(\\rightarrow\\) B<\/p>\n

\n

R : {(a,b) | (a,b) \\(\\in\\) A \\(\\times\\) B and a R b}<\/p>\n<\/blockquote>\n

Let A and B be two non empty sets and R : A \\(\\rightarrow\\) B be a relation such that R : {(a,b) | (a,b) \\(\\in\\) R a \\(\\in\\) A and b \\(\\in\\) B}<\/p>\n

(i) ‘b’ is called image of ‘a’ under R.<\/p>\n

(ii) ‘a’ is called pre-image of ‘b’ under R.<\/p>\n

(iii) Domain of R : Collection of all elements of A which has a image in B.<\/p>\n

(iv) Range of R : Collection of all elements of B which has a pre-image in A.<\/p>\n

Note :<\/strong><\/p>\n

(1) It is not necessary that each and every element of set A has a image in set B and each and every element of set B has preimage in set A.<\/p>\n

(2) Elements of set A having image in B is not necessarily unique.<\/p>\n

(3) Basically relation is the number of subsets of A \\(\\times\\) B<\/p>\n

Number of relations = no. of ways of selecting a non-zero subset of A \\(\\times\\) B<\/p>\n

= \\(^{mn}C_1\\)+ \\(^{mn}C_2\\) + …….. + \\(^{mn}C_{mn}\\) = \\(2^{mn} – 1\\)<\/p>\n

Total number of relation = \\(2^{mn}\\)(including void relation)<\/p>\n\n\n

Example : <\/span> If A = {1, 3, 5, 7}, B = {2, 4, 6, 8}
\n\t\t Relation is aRb \\(\\implies\\) a > b, a \\(\\in\\) A, a \\(\\in\\) B<\/p>\n

Solution : <\/span>R = {(3, 2), (5, 2), (5, 4), (7, 2), (7, 4), (7, 6)}
\n\t\t Domain = {3, 5, 7}
\n\t\t Range = {2, 4, 6}

<\/p>\n\n\n\n

\n
Next – Types of Relations in Math \u2013 Reflexive, Symmetric<\/a><\/div>\n<\/div>\n\n\n\n
<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn what is cartesian product of sets and what is relation and inverse relation with example. Let’s begin – Cartesian Product of Sets The cartesian product of two sets A, B is a non-void set of all ordered pair (a,b), where a \\(\\in\\) A and b \\(\\in\\) B. This is denoted by …<\/p>\n

What is Cartesian Product of Sets – Definition and Example<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[17],"tags":[576,577],"yoast_head":"\nWhat is Cartesian Product of Sets - Definition and Example - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn what is cartesian product of sets and what is relation with example.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/cartesian-product-of-sets\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"What is Cartesian Product of Sets - 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