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So, we have another kind of very interesting piece of the course right now. We're going to continue to talk about genetics, except now we're going to talk about the genetics of diploid organisms, which apart from bacteria, most of the organisms including us are diploid. They have more than one copy of each chromosome, and so we'll go through a segment on this, and also talk about mitosis and meiosis, the central processes of cell division and the segregation of genetic material that underlie life as we know it. And then, were going to charge into a session of recombinant DNA, and some of these technologies, PCR and various things that you see in the newspapers all the time. And then, I'll finish up with the session on the immune system, which a few of you thought was surprising that bacteria recombine. I'll tell you in that system it will feel like science fiction relative to what I've told you up to now. It's an absolutely amazing system. So, ...

Equivalence relations are another kind of binary relation on a set which play a crucial role in mathematics and in computer science in particular. And they can also be explained both in terms of digraphs and in terms of axioms. So let's begin with a digraph explanation of an equivalence relation. And the kind of relation that's an equivalence relation is the relation of there being a walk in both directions between two vertices. So if there's a walk between vertex u and vertex v and conversely there's a walk from vertex v back to vertex u, then u and v are said to be strongly connected, and strongly connected is going to be an example of an equivalence relation. So in terms of the walk relation, including 0-length walks, the relation we're talking about is u G* v and v G* u. Now, as a property of relations, this has a name. It's called symmetry. So a relation R on a set A is symmetric if and only if a R b implies b R a, and the first remark is that...