Show all work and explain each steps Given S012345 find the

Show all work and explain each steps

Given S={0,1,2,3,4,5}, find the partition induced by the equivalence relation R where R={(0,0),(0,4),(1,1),(1,3),(4,5),(0,5),(5,4),(5,0),(5,5),(2,2),(3,1),(3,3),(4,0),(4,4)}. Explain.

Solution

R is an equivalence relation on S if R is reflexive, symmetric, and transitive.

Relation R is reflexive if xRx for every x S. That is, R is reflexive if x S, xRx

R={(0,0),(0,4),(1,1),(1,3),(4,5),(0,5),(5,4),(5,0),(5,5),(2,2),(3,1),(3,3),(4,0),(4,4)} and S={0,1,2,3,4,5}

Relation R is symmetric if xR y implies yRx for all x, y S That is, R is symmetric if x, y S, xR y yRx

Relation R is transitive if whenever xR y and yRz, then also xRz. That is, R is transitive if x, y, z S, ¡ (xR y)(yRz) ¢ xRz.