Find whether the following partially ordered sets are lattices, whether they have a zero and unit elements. and whether they are complemented lattices:
(a) The set of all subsets of a finite set A, partially ordered by inclusion $\subseteq.$
I feel that it is a lattice because every 2 elements has an infmium which is the smallest of the two, and every two elements has a suprimium which is the largest of the two. and I feel that the zero element is the set $\phi$ and the unit element is the set A. But I do not know if every element has a complement? and if so could anyone give me an example of an element and its complement.
(b)The set of all ordered pairs $(a,b)$ of real numbers, partially ordered by $\leq$, where: $(a_{1},b_{1})$$\leq$$(a_{2},b_{2}$) iff $(a_{1} < a_{2})$ or ($(a_{1} = a_{2}$ & $b_{1}$$\leq$$\ b_{2}$).
I feel it is a lattice because every two elements has an infimium which is (min $(a_{1},a_{2})$, min $(b_{1},b_{2})$) and every two elements has a supremium which is (max $(a_{1},a_{2})$, max $(b_{1},b_{2}).$)
thanks.
You are right about the zero and unit elements being $\emptyset$ and $A$, respectively. However, the infimum is the intersection of two elements, and the supremum is the their union. The complement of a subset $B$ of $A$ is simply $A \setminus B$.
Say $A = \{0,1,2,3\}$, $B = \{1,3\}$ and $C = \{2,3\}$. Then the infimum of $B$ and $C$ is $\{3\}$, their supremum is $\{1,2,3\}$, and the complement $B^c$ of $B$ is $\{0,2\}$. You can verify that $B \cap B^c = \emptyset$ and $B \cup B^c = A$ as required by the definition of complement.
All in all, the set of all subsets of a finite set $A$, partially ordered by inclusion, is a complemented lattice. It is also distributive, as you may verify, which means that it is a Boolean algebra. In fact, all Boolean algebras of cardinality equal to $2^{|A|}$ are isomorphic to it. (See here.)