Question

Give an example of each of the following. If you use a familiar example that was covered in the class tutorials, you can simply cite the example with no proof. a. A set S and a partial ordering on S that is not a linear order. b. A set T and a linear ordering on T that is not a well ordering.

Answer 1

solution oiven that s uport can zet x = {0, 17 and s= o(x ) = { 0, {0} $17 {013] anch we give s one partial ardor no thad is the subset propertity clearly subset property is a partial order. the as į is reflective CA SA VA EPCXJ) cis anti symmetric Cif ASB and BCA then E is . A =B VA BEP(x)) transitive CACB and BCC » ACC & A, B, CEP(x)) but s is not a total order because {o y {12 Is the not comparable (as {0} & {1} and ... { 13% {0}} Hence, c is partial anden but not linear ondor

(br. we show if t=Co, a) and the linear onder A "L" an T is not an well ordering clearly ? " is a linear anden ont as ata & RET Coreflective) ass and be a j aib Cantisymmetric) all and b z c = arc (Toransmitive) and any two elements in T are comparable (by low of trichotomy) but we show it doesn't have a minimal element ondegi "L act or is a minimal element on the contorary of t under e B rooce oso let us now take see clearly f and en so SGT and ser or... gal

which contradicts the minimal element on least element Hence L is not well ordering Thank you