Question

In: Advanced Math

Prove that each element in Pentagon D5 has a unique inverse under the binary operation. D5={AF,...

Prove that each element in Pentagon D5 has a unique inverse under the binary operation.

D5={AF, BF, CF, DF, EF,0,72,144,216,288}

Expert Solution

Solution:

Set of symmetries of an regular pentagon with vertices levelled as in anticlockwise order. There are rotations ( ) where means rotation about degrees and reflections () where means reflection about the vertices .

Observe that

We can consider each rotations and reflections as a bijective function. For example in ,If we take an reflection say ,then
rotation about zero degree. this means there is no change in the original figure, so is treated as an identity function.
since we saw that each element of can be treated as a bijective function, so it is obvious that we can take binary operation as composition of function.
_{[This example shows you
how the operation works.]}

We can prove the statement through Cayley Table.

From the Cayley table one can say that the inverse element of that is , similarly and . Here from the Cayley table we conclude that each element of has unique inverse element. hence proved.

Colby Messinger answered 2 years ago

a. Prove that for any vector space, if an inverse exists, then it must be unique....

a. Prove that for any vector space, if an inverse exists, then it must be unique. b. Prove that the additive inverse of the additive inverse will be the original vector. c. Prove that the only way for the magnitude of a vector to be zero is if in fact the vector is the zero vector.

Prove that every complete lattice has a unique maximal element. (ii) Give an example of an...

Prove that every complete lattice has a unique maximal element. (ii) Give an example of an infinite chain complete poset with no unique maximal element. (iii) Prove that any closed interval on R ([a, b]) with the usual order (≤) is a complete lattice (you may assume the properties of R that you assume in Calculus class). (iv) Say that a poset is almost chain complete if every nonempty chain has an l.u.b. Give an example of an almost chain...

Prove that if a finite group G has a unique subgroup of order m for each...

Prove that if a finite group G has a unique subgroup of order m for each divisor m of the order n of G then G is cyclic.

Each element has a unique atomic number, which is listed on the periodic table above the...

Each element has a unique atomic number, which is listed on the periodic table above the chemical symbol for the element. The atomic number is equal to the charge on the nucleus. The atomic number also equals the number of protons in the nucleus and the number of electrons in the neutral atom of the element. For example, Fluorine (F) has the atomic number 9, which means that its atoms contain 9 protons and 9 electrons. Zinc (Zn) has the...

Experiments show that each element in the periodic table has a unique set of spectral lines....

Experiments show that each element in the periodic table has a unique set of spectral lines. What is the best explanation for this observation? Each atom has a unique nucleus. The number of electrons within each atom is unique. Each element has a unique set of interactions between its electrons and its protons. The motion of the electrons within each atom is unique. Each atom has a unique set of energy levels.

For each of the following determine whether ∗ is a binary operation on R. If so,...

For each of the following determine whether ∗ is a binary operation on R. If so, determine whether or not ∗ is associative, commutative, has an identity element, and has inverse elements. (a) a ∗ b = (ab) / (a+b+1) (b) a ∗ b = a + b + k where k ∈ Z (c) a ln(b) on {x ∈ R | x > 0}

Consider the mathematical system with the single element {1} under the operation of multiplication. Answer and...

Consider the mathematical system with the single element {1} under the operation of multiplication. Answer and explain the following: a) Is the system closed? b) Does the system have an identity element? c) Does 1 have an inverse? d) Does the associative property hold? e) Does the commutative property hold? f) Is {1} under the operation of multiplication a commutative group?

Prove that Z/nZ is a group under the binary operator "+" for every n in positive...

Prove that Z/nZ is a group under the binary operator "+" for every n in positive Z, where Z is the set of integers.

Suppose that a set G has a binary operation on it that has the following properties:...

Suppose that a set G has a binary operation on it that has the following properties: 1. The operation ◦ is associative, that is: for all a,b,c ∈ G, a◦(b◦c)=(a◦b)◦c 2. There is a right identity, e: For all a∈G a◦e=a 3. Every element has a right inverse: For all a∈G there is a^-1 such that a◦a^-1=e Prove that this operation makes G a group. You must show that the right inverse of each element is a left inverse and...

Using a system of equations, prove that a 2x2 matrix A has a right inverse if...

Using a system of equations, prove that a 2x2 matrix A has a right inverse if and only if it has a left inverse.