90 2 Algebra
(a)e=e′;
(b)x∗y=x◦y, for everyx, y∈M;
(c)x∗y=y∗x, for everyx, y∈M.272.Consider a setSand a binary operation∗onSsuch thatx∗(y∗x)=yfor all
x, yinS. Prove that each of the equationsa∗x=bandx∗a=bhas a unique
solution inS.
273.On a setMan operation∗is given satisfying the properties
(i) there exists an elemente∈Msuch thatx∗e=xfor allx∈M;
(ii)(x∗y)∗z=(z∗x)∗yfor allx, y, z∈M.
Prove that the operation∗ isboth associative and commutative.
274.Prove or disprove the following statement: IfFis a finite set with two or more
elements, then there exists a binary operation∗onFsuch that for allx, y, zinF,
(i)x∗z=y∗zimpliesx=y(right cancellation holds), and
(ii)x∗(y∗z) =(x∗y)∗z(no case of associativity holds).
275.Let∗be an associative binary operation on a setSsatisfyinga∗b=b∗aonly if
a=b. Prove thata∗(b∗c)=a∗cfor alla, b, c∈S. Give an example of such
an operation.
276.LetSbe a set and∗a binary operation onSsatisfying the laws
(i)x∗(x∗y)=yfor allx, y∈S,
(ii)(y∗x)∗x =yforallx, yinS.
Show that∗is commutative but not necessarily associative.
277.Let∗be a binary operation on the setQof rational numbers that is associative and
commutative and satisfies 0∗ 0 =0 and(a+c)∗(b+c)=a∗b+cfor alla, b, c∈Q.
Prove that eithera∗b=max(a, b)for alla, b∈Q,ora∗b=min(a, b)for all
a, b∈Q.
2.4.2 Groups.................................................
Definition.A group is a set of transformations (of some space) that contains the identity
transformation and is closed under composition and under the operation of taking the
inverse.
The isometries of the plane, the permutations of a set, the continuous bijections on a
closed bounded interval all form groups.
There is a more abstract, and apparently more general definition, which calls a group
a setGendowed with a binary operation·that satisfies
(i) (associativity)x(yz)=(xy)zfor allx, y, z∈S;