Problem 6

MIME-Version: 1.0 Content-Type: multipart/related; boundary="----=_NextPart_01C785BA.55E068D0" This document is a Single File Web Page, also known as a Web Archive file. If you are seeing this message, your browser or editor doesn't support Web Archive files. Please download a browser that supports Web Archive, such as Microsoft Internet Explorer. ------=_NextPart_01C785BA.55E068D0 Content-Location: file:///C:/753AC876/2007prob6.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Problem 6

Pro= blem 6

The functions _{and _{permute the=
order
of a set of three elements as follows:}}

_{denotes the=
usual
composition of two functions:}

a) What is the ou= tput of _?

b) What is the id= entity element, in terms of only _{, _{, or a composition of _{and _?}}}

c) List all of the possible one-to-one functions that permute the set of three elements in ter= ms of only _{, _{, or a composition of _{and _.}}}

d) What is the in= verse function of _{in terms of=
only _{, _{, or a composition of _{and _?}}}}

[Problem submitted b= y L. Zambrano, LACC Professor of Mathematics.]<= /span>

Solution for Prob= lem 6:

a) _{=3D_\}

b) _OR _.

_{: Since _{switches the order of the first two elements, applying twice would
return elements to their original order (and hence, an identity function): =
₌}}

OR, _{: since _{“shif=
ts
elements one to the right”, applying _{three times=
would
return all three elements to their original order.}}}

c) There must be = six, since there are only 3! (factorial) =3D 6 possible permutations of a set of= three elements. Thus, in addition t= o the four identified above ( _{, _{, _{, and the identity _{), the two others are&nb=
sp;
_{and _.}}}}}

d) _{, since _{Since=
the
input _{and the out=
put _{are identical, _{must be the=
inverse
function of_{, since the composition of one with the other acts as the
identity.}}<=
/span>}}}}

(Recall, for any function_{, _{is the inve=
rse of
_{if and only=
if _{for each x in the domain of _{, and vice versa…)}}}}}

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