SUNY Geneseo Department of Mathematics
Monday, April 23
Math 239 01
Spring 2018
Prof. Doug Baldwin
“Particle Swarm Optimization”
Prof. Ahmad Almomani, SUNY Geneseo
Thursday, April 26, 2:30 PM
Newton 203
Extra credit as usual for roughly a paragraph on your connections to or reactions to the talk.
Thursday, May 3, 12:00 noon to 2:30 PM, in our regular room.
Comprehensive, but with an emphasis on material since the 2nd hour exam (e.g., induction, sets, functions, relations, infinite sets).
Designed to be 2 to 2 1/2 times as long as the hour exams, but you have 3 times as much time.
Rules and format otherwise similar to hour exams, especially the open-references rule.
I’ll provide a sample from a past semester.
I’ll bring donuts and cider.
Would you like one, probably on study day (Wednesday May 2)?
If so, what time and for how long?
Conclusion: a review session would be helpful, for around an hour. I’ll look into scheduling one and let you know what I get.
Now under way; you should have gotten an email about filling them out online.
I will see results (anonymously and after the end of the semester), and they are useful to me in improving courses for the future. So please fill them out.
Section 7.2 (and we’ll cover some of 7.3 today in class).
Find an example of an equivalence relation between Henry VIII’s relatives.
Same-husband-as / same-wife-as?
Same-father-as / same-mother-as?
Explain why your example really is an equivalence relation.
Same-husband-as and same-wife-as turn out not to be transitive when you have remarriages.
But same-father-as and same-mother-as are reflexive (you have the same mother as yourself), symmetric (if A and B have the same mother, then so do B and A) and transitive, as long as we’re talking about biological mothers rather than step-mothers (since everyone only has 1 mother, if A and B have the same mother, and B and C do too, that person must be the mother for all three of A, B, and C).
Given a relation ∼ on set A, the equivalence class for x, written [x], is the set of all elements y of A such that y ∼ x, i.e., the set of everything related to x.
What are the equivalence classes of same-father-as above?
Define a “congruent mod 7” relation on the integers to hold between a and b if and only if a ≡ b (mod 7).
Show that “congruent mod 7” is an equivalence relation.
Prove that a relation is an equivalence relation by showing that it has the 3 defining properties:
The relation is reflexive: For example, to show that a ≡ a (mod 7) note that a - a = 0, and 7 divides 0. (Recall that one definition of what it means to say that a ≡ b (mod n) is that n divides b - a.)
The relation is symmetric: In this example, if a ≡ b (mod 7), then b ≡ a (mod 7), because if 7 divides b - a then 7 also divides a - b.
The relation is transitive: In this case, show that if a ≡ b (mod 7) and b ≡ c (mod 7), then a ≡ c (mod 7).
The definition of equivalence relations as being symmetric, reflexive, and transitive.
Equivalence classes.
Exercises on equivalence relations and infinite sets, and a little bit on function inverses.
See the handout for details.
Finite and infinite sets.
Read section 9.1