23 votes 23 votes The number of binary relations on a set with $n$ elements is: $n^2$ $2^n$ $2^{n^2}$ None of the above Set Theory & Algebra gate1999 set-theory&algebra relations combinatory easy + – Kathleen asked Sep 23, 2014 • retagged Jun 27, 2017 by Arjun Kathleen 11.3k views answer comment Share Follow See 1 comment See all 1 1 comment reply Aalok8523 commented Jun 24, 2020 reply Follow Share Same question was asked in 1987, see below :- https://gateoverflow.in/82436/gate1987-9a 0 votes 0 votes Please log in or register to add a comment.
Best answer 29 votes 29 votes Answer: $C$ In a binary relation two elements are chosen from the set. So, with $n$ elements $n^2$ pairings are possible. Now, a relation can be any subset of these $n^2$ pairings and thus we get $2^{n^2}$ binary relations. Rajarshi Sarkar answered Jun 24, 2015 • edited Jun 11, 2018 by Milicevic3306 Rajarshi Sarkar comment Share Follow See all 3 Comments See all 3 3 Comments reply val_pro20 commented Jan 12, 2019 reply Follow Share @shaik could you pls explain more 2 votes 2 votes pritishc commented Dec 22, 2019 reply Follow Share Think of the boolean matrix of the relation, where an element is related to another element if the matrix entry for that pair is 1. If unrelated, 0. There are $n^2$ entries in this matrix. Each entry can be $1$ or $0$. Therefore there are $2^{n^{2}}$ possible choices and that many relations. 1 votes 1 votes Sanjay Sharma commented May 6, 2020 reply Follow Share what if binary word is removed 0 votes 0 votes Please log in or register to add a comment.
0 votes 0 votes max number of elements in a binary relation on a set of n elements = n x n = n^2 therefore number of binary relations= $2^{n^2}$ akshay_123 answered Sep 2, 2023 • edited Nov 25, 2023 by Hira Thakur akshay_123 comment Share Follow See all 0 reply Please log in or register to add a comment.