696
Modeling Arithmetic, Computation, and Languages
tHeoRem on thE uniquEnEss of invErsEs in A GrouP
For each x in a group 3G, # 4, x
−1
is unique.
PrACtiCe 16 Prove the preceding theorem. (Hint: Assume two inverses for x, namely y and z, and let
i be the identity. Then y = y # i = y # (x # z) = c .)
PrACtiCe 17 Write 10 as 7 + 12 3 and use the theorem on the inverse of a product to find (10)
−1
in the
group 3Z 12 , + 12 4.
If x and y belong to a group 3G, # 4, then x # y belongs to G and must have an
inverse element in G. Naturally, we expect that inverse to have some connection
with x
−1
and y
−1
, which we know exist in G. We can show that (x # y)
−1
= y
−1 # x
−1
;
thus the inverse of a product is the product of the inverses in reverse order.
Proof: We will show that y
−1 # x
−1
has the two properties required of (x # y)
−1
.
Then, because inverses are unique, ( y
−1 # x
−1
) must be (x # y)
−1
.
(x # y) # ( y
−1 # x
−1
) = x # ( y # y
−1
) # x
−1
= x # i # x
−1
= x # x
−1
= i
Similarly, ( y
−1 # x
−1
) # (x # y) = i. Notice how associativity and the meaning of i
and inverses all come into play in this proof. End of Proof.
ReminDeR
If it walks like a duck …
We know that many familiar number systems such as 3Z, + 4 and 3R, + 4 are
groups. We make use of group properties when we do arithmetic or algebra in
these systems. In 3Z, + 4, for example, if we see the equation x + 5 = y + 5, we
conclude that x = y. We are making use of the right cancellation law, which, we
will soon see, holds in any group.
Definition CAnCELLAtion LAws
A set S with a binary operation # satisfies the right cancellation law if for
x, y, z [ S, x # z = y # z implies x = y. It satisfies the left cancellation law if
z # x = z # y implies x = y.
tHeoRem on thE invErsE of A ProDuCt
For x and y members of a group 3G, # 4, (x # y)
−1
= y
−1 # x
−1
.
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