Download 541Classwork Day 2

541 Day 2 Question(s) – O – the Day: Suppose you have a set of elements, S, and an operation (called “multiplication”) that allows you to multiply any pair of elements and get another element from that set. Further suppose that every element can be “generated” by two elements, A and B, in the sense that any element can be obtained by multiplying these two elements by themselves or each other (perhaps many times). Finally suppose that A2 = I and B4 = I, where I is an identity element (the product of any element with I in any order just results in that element). Question: What can be said about how many elements there are in S? Answer (So Far): We Can Say Nothing! [Note that this answer uses things that are discussed below, but these are fairly basic so I’m just going to answer this here.] If this is all we know about the operation then we don’t know anything. If we know the operation is associative then we are in better shape because then we can regroup elements however we want to help us simplify. For instance associativity tells us that A (AB) is really the same as (AA) B, and since AA = A2 = I. This combo is not a new element, it is just B in disguise. Note that associativity also allows us to unambiguously multiply more than two things together. If we know that AB = BA then we DO know enough to answer the question definitively. Given any combo, we can regroup and then commute the elements so that we have all the A’s in a row followed by all the B’s. Then we can reduce down to AnBm where n can be 0 or 1 and m can be 0, 1, 2, or 3. Try it for this example: A2B3AA3 B7. This means there are at most 8 elements. [Fun question: We only said AB = BA. Does this mean the operation is commutative?] We still have left the job of figuring out how to handle this question if we don’t have AB = BA. 1. We looked at the course website and went over the tweaks to the syllabus. If you missed it, take yourself on a survey of the website and read over the new version of the syllabus. 2. What is a binary operation? a. Informally: Two things (from the same set) come in and one thing (from that same set) comes out. b. Formally: An operation ♦on a set S is a function from SxS to S. Note that this definition requires closure! 3. A couple of important properties: a. Commutative property: An binary operation ◊ on a set S is commutative if… ∀ a, b in S, a ◊ b = b ◊ a. b. Associative property: An binary operation ◊ on a set S is associative if… ∀ a, b, c in S, (a ◊ b) ◊ c= a ◊ (b ◊ c). 4. Examples (Here are some that I came with that are not in Section 2 of the book. Test them!): a. Give an example of a binary operation that is commutative and associative. Set is the set of diagonal 2x2 matrices. Both addition and multiplication are associative and commutative. Set of linear transformations from R to R (i.e. functions of the form f(x) = mx). Composition is both associative and commutative. b. Give an example of a binary operation that is neither commutative nor associative. Set is the nonzero rational numbers. Operation is division. Set is the positive integers. Operation is a◊b = ab. [None of these has an identity element. Look up the Octonions on the interwebs for an example that does. You can also cleverly use the above examples to include an identity element. Simply add a new element, call it “Bob” and extend the previous operation by saying that Bob combined with any other element returns that element. Now you still are non‐commutative and non‐associative, but now you have an identity element.] c. Give an example of a binary operation that is commutative but not associative. Set is the real numbers. Operation is taking the average: a◊b = (a + b)/2. d. Give an example of a binary operation that is associative but not commutative. Easy! Pretty much all of your function composition examples will have this. 5. Prove that function composition is always associative [Closed Book] You are supposed to try to prove this one using only the definition of function composition (without peeking in the book).