Notes also available as PDF.
Structure of the upcoming test:
Pólya’s principles:
This is not a simple 1234 recipe. Understanding the problem may include playing with little plans, or trying to carry out a plan may lead you back to trying to understand the problem.
Read the whole problem.
Read all of the problem.
One comment about the homeworks: Most people answer only part of any given problem.
To indicate an answer clearly to someone else (like me), you need to know what the answer is.
Rephrasing the problem may help you remember solution methods.
This is close to devising a plan. Sometimes you may stumble upon an answer.
Examples may help find relationships. The relationships may help you decide on a plan. Mathematics is about relationships between different entities; symbolic mathematics helps abstract away the entities themselves.
Sometimes plans are “trivial,” or so simple it seems pointless to make them specific. But write it out anyways. Often the act of putting a plan into words helps find flaws in the plan.
Try to devise a plan that you can check along the way. The earlier you detect a problem, the easier you can deal with it.
Some plans we’ve considered:
Try a few combinations of the data. See what falls out. This is good for finding relationships and understanding the problem.
If you know the answer lies in some range, you can search that range systematically by building a list.
When trying examples, keep an eye open for patterns. Sometimes the patterns lead directly to a solution, and sometimes they help to break a problem into smaller pieces.
Be sure to understand what results depend on which data. Look for dependencies in the problem. Sometimes pushing the data you have through all the dependencies will break the problem into simpler subproblems.
Attention to detail is critial here.
When building a list, be sure to carry out a welldefined procedure. Or when looking for patterns, be systematic in the examples you try. Don’t jump around randomly.
Can you check your result? Sometimes trying to check reveals new relationships that could lead to a better solution.
Think about how your solution could help with other problems.
You can write a set by listing its entries, \{1,2,3,4\}, or through set builder notation, \{x\kern 1.66702pt \kern 1.66702pt x\text{ is a positive integer},x < 5\}.
Note that 1 \{1,2\} and \{1\} \{\{1\},\{2\}\}, but \{1\}∉\{1,2\}.
One implication is that ∅⊂ A for all sets A. This statement is vacuously true.
Here \{1\} ⊂\{1,2\} and \{1\}⊄\{\{1\},\{2\}\}.
You can write the result of multiple operations in setbuilder notation,
The cardinality of a set is a count of its elements. So \{a,b,c\} = 3. An infinite set like the set of all positive integers has no simple cardinality.
To sets are equivalent, or A ~ B, if there is a onetoone correspondence between the two sets. With such a correspondence, each element of one set is matched with exactly one element of the other set.
For finite sets, the two sets must be the same size. If they weren’t, there would be at least one unmatched element.
For finite sets, establishing a onetoone correspondence is straightforward. You just need to list which entries correspond. For \{1,2,3\} and \{a,b,c\}, such a list might be
1 ↔ c,\quad 2 ↔ a,\quad 3 ↔ b.

Or you could provide a formula. For A = \{1,2,3\} and B = \{2,4,6\}, a mapping could be a A ↔ f(a) B where
f(a) = 2a.

For infinite sets, a formula is the most straightforward way. Consider the extending the above sets to be infinite. For sets A = \{1,2,3,\mathop{\mathop{…}}\} and B = \{2,4,6,\mathop{\mathop{…}}\}. the same mapping as before shows they are equivalent. For the sets A = \{1,2,3,\mathop{\mathop{…}}\} and B = \{2,3,4,\mathop{\mathop{…}}\}, a mapping function would be f(a) = a + 1.
There are multiple ways to illustrate addition and multiplication.
For addition, one illustration is “piles of rocks” or symbols. This is somewhat like set unions, except we assume every element of each set is unique. Equivalently, we assume all sets are disjoint. Then A ∩ B = ∅ = 0, so A ∪ B = A + BA ∩ BA + B.
Another illustration is the number line. Each number is represented by a length:
0  ∙ 
1  ∙—∙ 
2  ∙—∙—∙ 
Then addition appends lines:
1+2  ∙—∙ + ∙—∙—∙  ∙—∙—∙—∙  3 
This makes the commutative and associative properties fairly obvious; the line has the same final length regardless.
This is essentially the same as Peano arithmetic covered earlier, except addition in Peano arithmetic moves a bar from one number to the other until the sum is reached.
Multiplication often is best illustrated by areas or volumes.