answer in detail please Find the elements in the relation \"have the same remain

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answer in detail please

Find the elements in the relation "have the same remainder when divided by 7" if the relation is defined on X = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}. Also find the equivalence classes of this equivalence relation. Define a binary relation fionMasfollows: R = {(x,y) R times R: cos(x) = cos(y)} Prove that R is an equivalence relation, and determine its equivalence classes. State whether or not each of the following relations defined on X the relations is not a partial order, state why not X = {1,2,3,4} is a partial order. If any of relations is not a partial order, state why not. Are any of the relations in problem 3 linear orders? Let X = {a, b, c, d, e, f] Define a partial order R on X, to be as represented in the following diagram: This diagram is read from the bottom up, that is, the elements at the bottom are lowest in the ordering and the elements at the top are highest in the ordering. a Identify any minimal, minimum, maximal and maximum elements in this partial order.

Explanation / Answer

1)there are 7 equivalence classes,

[0]class={7,14} remainder:0

[1]class={1,8,15} remainder:1

[2]class={2,9} remainder:2

[3]class={3,10} remainder:3

[4]class={4,11} remainder:4

[5]class={5,12} remainder:5

[6]class={6,13} remainder:6

each class forms an equivalence relation

2)for equivalence we need to prove reflexive,symmetric,trnasitive

Reflexive:

consider x belongs to real numbers then x Related to x,since cosx=cosy

therefore R is reflexive

for symmetric

consider x,y belonging to real numbers,

given x related to y then cos(x)=cos(y)

then y related to x since cos(y)=cos(x)

therefore R is symmetric

for transitive :

consider x,y,z belonging to real numbers,

given x related to y then cos(x)=cos(y)

then y related to z since cos(y)=cos(z)

implies x related to z since cos(x)=cos(z)

therefore R is equivalance relation

equivalence classes are {x+2*n*pi} where n is integer

for different x belongs to 2*pi we get different classes

3)

a)it is not partial ordered set since it satisfies reflexive property

b)it is not partial ordered set since it does not satisfy transitivity

c)it is a partial set

4)yes ,{(1,1),(1,2),(1,3),(1,4),(2,2),(2,3),(2,4),(3,3),(3,4),(4,4)} is linearly ordered

5)maximum element : a

minimum elements are: d,e,f

minimal elements are:d,e,f

maximal element : a

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