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A cyclic set of order three, under multiplication, consists of three element, i, x, and x^2 such that x^3 = i where i is the identity.
For the set to be a group it must satisfy four axioms: closure, associativity, identity and invertibility. i*i = i, i*x = x, i*x^2 = x^2,x*i = x, x*x = x^2, x*x^2 = x^3 = ix^2*i = x^2, x^2*x = x^3 = ix^2*x^2 = x^4 = x^3*x = i*x = x Since each of the elements on the right hand side belongs to the set, closure is established.
It can, similarly be shown that the elements of the set satisfy the associative property.
As can be seen from the entries for closure, i is the identity.
Also, the inverse of i is i,the inverse of x is x^2 andthe inverse of x^2 is xand therefore, the set has invertibiity. It is, therefore a group.
What else can I help you with?
Is every abelian group is cyclic or not and why?
every abelian group is not cyclic. e.g, set of (Q,+) it is an abelian group but not cyclic.
How do we prove that a finite group G of order p prime is cyclic using Lagrange?
Lagrange theorem states that the order of any subgroup of a group G must divide order of the group G. If order p of the group G is prime the only divisors are 1 and p, therefore the only subgroups of G are {e} and G itself. Take any a not equal e. Then the set of all integer powers of a is by definition a cyclic subgroup of G, but the only subgroup of G with more then 1 element is G itself, therefore G is cyclic. QED.
What is finite and infinite cyclic group?
Normally, a cyclic group is defined as a set of numbers generated by repeated use of an operator on a single element which is called the generator and is denoted by g.If the operation is multiplicative then the elements are g0, g1, g2, ...Such a group may be finite or infinite. If for some integer k, gk = g0 then the cyclic group is finite, of order k. If there is no such k, then it is infinite - and is isomorphic to Z(integers) with the operation being addition.
Prove the demorgan's laws for crisp set theory?
prove the intersction for crisp set theory
What is the order of a group?
The order of a group is the same as its cardinality - i.e. the number of elements the set contains. The order of a particular element is the order of the (cyclic) group generated by that element - i.e. the order of the group {...a-4, a-3, a-2, a-1, e, a, a2, a3, a4...}. If these powers do not go on forever, it will have a finite order; otherwise the order will be infinite.
What is the difference between reversible and cyclic process?
A reversible process is one that can be undone with no change in entropy of the system and surroundings. A cyclic process is one that starts and ends at the same state, with the system going through a series of state changes. All reversible processes are cyclic, but not all cyclic processes are reversible.
What is the order of an element of a group?
The order of a group is the same as its cardinality - i.e. the number of elements the set contains. The order of a particular element is the order of the (cyclic) group generated by that element - i.e. the order of the group {...a-4, a-3, a-2, a-1, e, a, a2, a3, a4...}. If these powers do not go on forever, it will have a finite order; otherwise the order will be infinite.
What did Columbus set out to prove?
Columbus originally was determined to prove that
Are set of integers are cyclic under addition?
No. The additive identity, 0, is the only value such that A*0 = 0 for any non-zero element A of the set.
Prove that the boundary of a set is involved in that set only when this set is a closed set?
That is the definition of a closed set.
What is the order of grouping?
The order of a group is the same as its cardinality - i.e. the number of elements the set contains. The order of a particular element is the order of the (cyclic) group generated by that element - i.e. the order of the group {...a-4, a-3, a-2, a-1, e, a, a2, a3, a4...}. If these powers do not go on forever, it will have a finite order; otherwise the order will be infinite.
What describes a set of learned and mutually accepted rules that define all forms of communication for a group of people during a particular time period?
culture
