Question

Let S={1,2,3,4}, and define a partial ordering of P(S) (the power set of S), by: A⪯B if and only if A⊆B. Is this partial ordering in fact a total ordering (chain)? Why or why not?

Answer 1

For a set $S$ a partial ordering of $P(S)$ defined by: $A⪯B$ if and only if $A⊆B$, is not a total ordering because for the sets $A=\{1,2\}$ and $B=\{3,4\}$ it is not true that $A⊆B$ or $B⊆A.$