Question

Show that the number of binary operations on `{1, 2}`having 1 as identity and having 2 as the inverse of 2 is exactly one.

Answer 1

Given that `1` is identity.
`:. a**1 = 1**a = a`
When `a = 1, 1**1 = 1`
When `a = 2, 1**2 = 2**1 = 1`
Also given that `2` is the inverse of `2`.
`:. 2**2 = 2**2 = e`
Here, `e` is identity and equal to `1`.
`:. 2**2 = = 2**2 = 1`
Hence, there is only one binary operatio that saisfies the given conditions.