Let 𝐴 be the set of letters in the English alphabet.

Question

Let 𝐴 be the set of letters in the English alphabet. Show that the relation R = {(𝛼,𝛽)∈𝐴×𝐴: 𝛼 𝑐𝑜𝑚𝑒𝑠 𝑏𝑒𝑓𝑜𝑟𝑒 𝛽 𝑖𝑛 𝑎𝑙𝑝ℎ𝑎𝑏𝑒𝑡𝑖𝑐𝑎𝑙 𝑜𝑟𝑑𝑒𝑟} is a strict partial ordering on A

Answer 1

Transitive:

if, for (a,b) and (b,c) in R=AxA, a~b and b~c then

a<b and b<c, so that a preceeds b and b preceeds c;

therefore a<b<c and a preceeds c, so a~c

Hence (a,c) is in AxA=R

Asymmetric:

if for (a,b) in R=AxA, a~b, then

a< b, so that a preceeds b.

Then b>a, so a b does not proceed a.

Therefore (a,b) is asymmetric

[end of proof]