{"id":9806,"date":"2022-01-28T01:49:22","date_gmt":"2022-01-27T20:19:22","guid":{"rendered":"https:\/\/mathemerize.com\/?p=9806"},"modified":"2022-01-28T01:49:25","modified_gmt":"2022-01-27T20:19:25","slug":"one-one-function-injection-definition-and-examples","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/one-one-function-injection-definition-and-examples\/","title":{"rendered":"One One Function (Injection) – Definition and Examples"},"content":{"rendered":"

Here you will learn what is one one function with definition and examples.<\/p>\n

Let’s begin –<\/p>\n

One One Function Definition<\/h2>\n

Definition<\/strong> : A function f : A \\(\\rightarrow\\) B is said to be a one-one function or an injection if different elements of A have different images in B.<\/p>\n

\n

Thus, f : A \\(\\rightarrow\\) B is one-one<\/p>\n

\\(\\iff\\)  a \\(\\ne\\) b  \\(\\implies\\)  f(a) \\(\\ne\\) f(b) for all a, b \\(\\in\\) A<\/p>\n

\\(\\iff\\)  f(a) = f(b)  \\(\\implies\\)  a = b for all a, b \\(\\in\\) A<\/p>\n<\/blockquote>\n

Also Read<\/strong> : Types of Functions in Maths \u2013 Domain and Range<\/a><\/p>\n

Example<\/strong><\/span> : Let f : A \\(\\rightarrow\\) B and g : X \\(\\rightarrow\\) Y be two functions represented by the following diagrams :<\/p>\n

\"one\"one<\/p>\n

Solution<\/strong><\/span> : Clearly, f : A \\(\\rightarrow\\) B is a one one function. But, g : X \\(\\rightarrow\\) Y is not one-one because two different elements x1 and x3 have the same image under function g.<\/p>\n

Example<\/strong><\/span> : Let A = {1, 2, 3, 4}, B = {1, 2, 3, 4, 5, 6} and f : A \\(\\rightarrow\\) B be a function defined by f(x) = x + 2 for all x \\(\\in\\) A.<\/p>\n

Solution<\/strong><\/span> : We observe that f as a set of ordered pairs can be written as f = {(1, 3), (2, 4), (3, 5), (4, 6)}<\/p>\n

Clearly, different elements in A have different images under function f.<\/p>\n

So, f : A \\(\\rightarrow\\) B is an injection or one-one.<\/p>\n

Algorithm to Check for Injection <\/h2>\n
\n

1).  Take two arbitrary elements x, y (say) in the domain of f.<\/p>\n

2).  Put f(x) = f(y)<\/p>\n

3).  Solve f(x) = f(y). If it gives x = y only, then f : A \\(\\rightarrow\\) B is a one-one function (or an injection) otherwise not.<\/p>\n<\/blockquote>\n

Number of One One Function<\/h2>\n

If A and B are two sets having m and n elements respectively such that m \\(\\le\\) n, then the total number of one-one functions from A to B is<\/p>\n

\n

\\(^nC_m \\times m!\\)<\/p>\n

where m! is m factorial.<\/p>\n<\/blockquote>\n\n\n

<\/p>\n","protected":false},"excerpt":{"rendered":"

Here you will learn what is one one function with definition and examples. Let’s begin – One One Function Definition Definition : A function f : A \\(\\rightarrow\\) B is said to be a one-one function or an injection if different elements of A have different images in B. Thus, f : A \\(\\rightarrow\\) B …<\/p>\n

One One Function (Injection) – Definition and Examples<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","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":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[26],"tags":[359,833,831,832,366],"yoast_head":"\nOne One Function (Injection) - Definition and Examples<\/title>\n<meta name=\"description\" content=\"In this post you will learn what is one one function with definition and examples based on it.\" \/>\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\/one-one-function-injection-definition-and-examples\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"One One Function (Injection) - 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