MAT118 Review for Final

Material from Week 11

Understand and be able to reproduce the proof that $\sqrt{2}$ is irrational (if $\sqrt{2} = \frac{\displaystyle m} {\displaystyle n}$, where $m$ and $n$ are integers, then we can simplify the fraction if necessary to get $\sqrt{2} = \frac{\displaystyle p}{\displaystyle q}$ where $p$ and $q$ are integers with no common divisors; go from there to get a contradiction). Be able to replace $2$ by $5$, $7$ etc. (as in Mindscape 10 p.121).

Decimal expansions. Given a decimal that is periodic after some position (e.g. $27.6522222....$ or $768.2134343434...$ be able to represent that number as a fraction. (See p.128). Conversely, given any fraction, be able to use long division to show that its decimal expansion is periodic after some point. (See p.129).

Material from Week 12

Understand why a point picked at random on the number line has no chance of being a rational number.

Material from Week 13

Understand what it means for the elements of two sets (collections) to be in one-to-one correspondence. Understand that the operation of counting a collection of objects amounts to setting up a one-to-one correspondance between the elements of that collection and the elements $1, 2, 3, \dots, n$ of one of the canonical finite sets: $\{1\}$, $\{1, 2\}$, $\{1, 2, 3\}$, $\{1, 2, 3, 4\}$, etc. And the definition of cardinality for a finite collection: a finite collection has cardinality $n$ if its elements can be put in one-to-one correspondance with $\{1, 2, 3, \dots, n\}$.

Understand that we can extend the concept of cardinality to infinite sets by the definition: Two sets have the same cardinality if there is a one-to-one correspondance between their elements. (Understand that for finite sets this meshes with the earlier definition of cardinality).

Be able to explain why the set of counting numbers $\{1, 2, 3, \dots \}$ and the set of integers $\{\dots, -3, -2, -1, 0, 1, 2, \dots\}$ have the same cardinality. Be able to explain why the set of counting numbers $\{1, 2, 3, \dots\}$ and the set of rational numbers (numbers that can be written as $\frac{m}{n}$ where $m$ and $n$ are integers) have the same cardinality.

Be able to explain Cantor's Diagonal Argument showing that the set of points on the number line between $0$ and $1$ cannot be put in one-to-one correspondance with the set of counting numbers.

Material from Week 14

Understand the concepts of set and subset. Understand the empty set $\emptyset$. Given a set $S$ understand the definition of the power set $\cal{P}(S)$ as the set of all subsets of $S$. Understand that if $S$ is a finite set of cardinality $n$, then $\cal{P}(S)$ has cardinality $2^n$. Be able to write down all the elements of $\cal{P}(S)$ for some small set like $S=\{ \mbox{banana}, \mbox{papaya}, \mbox{grape}\}$. Understand "Cantor's Power Set Theorem" (p. 179): For any set $S$ the cardinality of $\cal{P}(S)$ is always strictly greater than the cardinality of $S$.

May 3, 2013