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.