We missed class on Tuesday due to the student demonstrations on Monday night.  And then the class was moved to Friday.  Sigh.

Friday, 5 October 2012: We went over the quizzes from last Wednesday, and built on some ideas about Z_n and modular arithmetic.  Then we discussed cardinalities of infinity, and bijections between certain infinite sets.

Readings:
Infinity is for Children - And Mathematicians!
Hotel Infinity
Wikipedia: Hilbert's Grand Hotel

 Suggested Problems:

1) It's a good time to try the challenge problem from the quiz-day last week!  First, compute {x^2 | x in Z_3}.  Then, show that if a, b, and c are integers such that a^2+b^2=c^2, then at least one of a, b, and c is divisible by 3.

Next, compute {x^2 | x in Z_2} and show that if a, b, and c are integers such that a^2+b^2=c^2, then at least one of a, b, and c is divisible by 2.

Next, do it again for Z_5.

Can this reasoning work for any other prime numbers?

2) Is there a bijection between the natural numbers and the rational numbers?  (Tool: If S and T are sets, and there is an injection from S to T and another injection from T to S, then T and S have the same cardinality.)

3) Is there a bijection between polynomials with positive integer coefficients and the natural numbers?

Name
Admission Number
Q1.1: How many injections are there from Z_12 to Z_11?
Q1.2: How many surjections are there from Z_6 to Z_15?
Q1.3: How many bijections are there from Z_13 to Z_8?
Q2: Let f:Z_7->Z_7 given by f(x)=x^6-1. Find {f(x) for x in Z_7}.
Q3: Let S be an infinite set. Remove every eighth element of S. Is S still infinite?
Q4.1: Is there a bijection from N to Z?
Q4.2: Is there a bijection from the even natural numbers to the odd integers?
Q5: Write the definition of an injective function.
 

One Response to SMA109 - Week 2: Functions and Infinity

  1. Kennedy Okoth Opere says:

    I am so much greatfull for the last lesson whereby it was perfectly done and I hope that it will be the same today. Thank you so much as I am preparing for the cat's and end semester exms.Good day

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Vi Hart - Hyperactive videos about beautiful math concepts. Snowdecahedron - A mathematical art installation. Tau - An alternative to pi. BBC Brief History of Math - A Documentary. John Baez - A maths superhero.
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