23 votes 23 votes How many one-to-one functions are there from a set $A$ with $n$ elements onto itself? Set Theory & Algebra gate1987 set-theory&algebra functions descriptive + – makhdoom ghaya asked Nov 14, 2016 recategorized Apr 22, 2021 by Lakshman Bhaiya makhdoom ghaya 4.2k views answer comment Share Follow See all 0 reply Please log in or register to add a comment.
Best answer 29 votes 29 votes There are $n!$ one to one function possible from a set of $n$ elements to itself. i.e., $P\binom{n}{n} = n!$ Prashant. answered Nov 14, 2016 edited Jun 8, 2018 by Arjun Prashant. comment Share Follow See all 2 Comments See all 2 2 Comments reply Vicky rix commented Sep 19, 2017 reply Follow Share All these n! functions will also be onto and so they all are bijections ... 6 votes 6 votes talha hashim commented Jan 1, 2019 reply Follow Share @vicky sir u are right 0 votes 0 votes Please log in or register to add a comment.
20 votes 20 votes f : A---> A ∣A∣ = n The first element of the domain has n choices for mapping,2nd element has (n-1) ,3rd element has (n-2) choices and so on. So, total number of one-to-one functions = n ⨉(n-1)⨉(n-2)⨉(n-3).........⨉1 = n! The correct answer is n! Warrior answered Aug 14, 2017 edited Nov 29, 2023 by Hira Thakur Warrior comment Share Follow See all 0 reply Please log in or register to add a comment.
0 votes 0 votes . akshay_123 answered Sep 24, 2023 akshay_123 comment Share Follow See all 0 reply Please log in or register to add a comment.