# Things Familiar and Less Familiar(algebra)

Things Familiar and Less Familiar(algebra)

Let S be a set having an operation ∗ that assigns an element an x b of S for any a, b ∈ S. Let us assume that the following two rules hold: If a, b are any objects in S, then an x b = a. If a, b are any objects in S, then a x b = b x a . Show that S can have at most one object.

Let S be a set having an operation * which assigns an element a*b of S for any a, b, elementof S.

Let us assume that the following two rules hold

(i) If a, b are any objects in S, then a*b = a

(ii) If a, b are any objects in S, then a*b = b*a

Take two elements a and b∈S. We have the following equalities due to the hypotheses:

a=a∗b=b∗a=b.

So a=b,

Since a and b where arbitrary we conclude S has at most one element.