Define a binary operation * on the set `A={1,2,3,4}`as `a*b=a b`(mod 5). Show that 1 is the identity for * andall elements of the set A are invertible with`2^(-1)=3`and `4^(-1)=4`

Define a binary operation * on the set `A={1,2,3,4}`as `a*b=a b`(mod 5

1.	Define a binary operation * on the set `A={1,2,3,4}`as `ab=a b`(mod 5). Show that 1 is the identity for andall elements of the set A are invertible with`2^(-1)=3`and `4^(-1)=4`
Answer» Here, `A = {1,2,3,4}.` And, `ab = ab mod 5.`So, if we take any two elements from `A`, Then we can define the given binary operation. For example,`11 = (11) mod 5 = 2` `22 = (22) mod 5 = 4` `2*3 = (23) mod 5 = 1` `4*4 = (44) mod 5 = 1` Similarly, we can define the given binary operation on each order pair of `A`. Please refer to video to see the complete table for this binary operation. Now, we will find the identity element of this relation. An identity element `e` in a relation is an element such that `ae = ea = a.` In this case, identity element is `1`. `:. e = 1.` Now, we know, if `ab = e`, then `a = b^-1`. Here, `23 = 1 => 2^-1 = 3`. Also, `44 = 1 => 4^-1 = 4`.