Supremum, infimum, powerset...

Question

Given the set $S = \{a, b, c, d\}$, and the powerset $P(S)$ of $S$.

The set-inclusion relationship is denoted with $⊆$. In the ordered set $(P(S), \subseteq)$:

$\sup_{P(S)}(\{a, c\}, \{a, d\}) = \{a,c,d\}$ ?

$\inf_{P(S)} (\{a, c\}, \{a, d\}) = \{\{a\},\{c\},\{d\}\}$ ?

Given $A = \{X \in P(S): | X | \le 2\}$. How many and what are the elements of $A$?

Please help me. What does $A = \{X \in P(S): | X | ≤ 2\}$ mean? Are my $\sup$ and $\inf$ correct?

Answer 1

$A$ is the family of elements of the power set, i.e. the family of subsets of $S$, that contain at most two letters. You should find all subsets of $S$ of this form and count how many they are.

Post scriptum: in general, if $|S| = n$, and if $A_k := \{X \in P(S)\ |\ |X| \leq k\}$, then $$|A_k| = \sum_{j=0}^k\binom n j$$