Exercise 4.1 (9.28 in the book).
Show that there is a formal proof in the first-order language with non-logical symbols (where is a unary function and are constants) of from and .
(Hint: first make an informal argument in the way you might for some other mathematics course you are studying. Then try to tighten up the arguments and write all the cases down. Then rewrite your proof in first order logic.)
Let be the first order language with single non-logical symbol (a binary relation).
(a) Write down statements of saying that is an equivalence relation.
(b) Write down a single statement of saying that there are exactly three -equivalence classes.
(c) Show how to write down infinitely many statements of that when taken together say that there are infinitely many -equivalence classes.
Let be the first order language with a single non-logical symbol (a binary relation). A simple graph is a nonempty set of vertices and a set of unordered pairs of vertices.
(a) Explain how an -structure might be regarded as a graph, and vice versa. What two extra conditions on -structures must be true for this graph to be a simple graph?
(b) Write down statements of saying that:
(i) the graph is complete (has every possible edge);
(ii) the graph has no cycle of length ;
(iii) vertices of are not connected by any path. (You will need to give infinitely many sentences for this one.)
A simple graph is bipartite if there is a partition into two disjoint sets such that all edges in join a vertex in with a vertex in . If is a simple graph and is an equivalence relation on then we define a graph by: for the -equivalence class of is denoted and is the set of -equivalence classes of ; and if and only if for some and . (Note that this construction may create loops, i.e. edges from a vertex to itself, which is also a cycle of length 1.) Suppose is a simple graph with no cycles of odd length i.e. no sequence of vertices with for all and also .
(a) Let be the set of equivalence relations on such that has no cycles of odd length. Show that is nonempty.
(b) Define on by letting if and only if implies for all . Show that is a poset with the Zorn property.
(c) Let be maximal. Show that divides into either one or two equivalence classes and the case of only one class is only possible is the original graph had no edges. (Hint: two equivalence classes can be merged provided (a) there is no path of odd length between them and (b) provided there is no third class with paths of from to and from to of different parity. It follows that any two classes with an even length path between them can be merged, and if two classes cannot be merged but have no path between, then there is another class with an even length path from one of them.)
(d) Deduce that any graph with no odd cycles is bipartite. Prova a converse too, and hence explain how to write down infinitely many sentences of saying that a graph is bipartite.