8/26 | 28 | 31 | 9/2 | 4 | 7 no class | 9 | 11 | 14 |
We began a discussion of sets by considering two examples: S={1,3,5,7}
and T=(red,blue, yellow} and the issue of what it means for a set to be
"well defined." In particular for a set to be well defined one must be
able to determine whether something is a member or not of the set. Thus
3 is a member of S, 2.999 is not a member of S, and 2.9999... is a member.
( 2.999... is another way to denote the number 3, so we distinguish the
number 3 from the way it is represented with numberal or using decimals.)
[ Let A = 2.99... . Then 10A= 29.9..., and hence 9A=10A-A=27. So A
= 27/9 = 3. ]
Next time: More on sets.
Here are some of the figures we used to think about the problem. (still
to be done....)
We spent a little time at the end of class on the nature of work that uses sets... in particular the importance of recognizing the context (or universe) in which a disucussion using sets transpires. The contexts of mathematics related to numbers were connected to various levels of school work.
counting numbers | primary grades |
fractions | elementary grades |
negative numbers | pre-algebra |
rational numbers | pre-algebra |
real numbers
sqrt(2) and pi |
geometry and algebra |
imaginary and complex numbers | algebra |
We also distinguished numbers from the notations used to represent them, e.g., which is larger 3 or 5 ? and the symbol for pi doesn't tell you anything about the nature of this number, while sqrt(2) does indicate that this number when squared equals 2.
Next class: More on sets! and TA 011.
We continued the discussion of sets and the notation. Lists, verbal
descriptives, and list patterns were illustrated with examples showing
again the need to be aware of the context of a universal set and to be
explicit about patterns that are sometimes presumed. Set equality was noted
as depending on the ability to identify set membership and to be able to
justify that the elements of each set are members of the other, i.e.,
Def'n: If A and B are sets, we say the sets A and B
are equal, A=B, when every element of A is an element of B and
every element of B is an element of A.
We then turned our attention to some elementary set operations: Union, intersections, subtraction, and complement.Each of these was illustrated with an example where the sets were given by lists.
Next time: More on sets... and a beginning to some more details on the nature of proof.
Back to a discussion of sets, we introduced the relation between
sets of one set being a subset of another, and saw how we could use the
definition to examine when this relation is not true and when it is. [I
have more to write here!] An example of some sets given by lists, and
one where the sets are given by decriptions using inequalities.
At the end of class we started further discusion of Cartesian products
of sets. In particular we discussed how to visualize products using horizontal
and vertical features to locate points in a picture of a product.
Here is an example:
A | AxB | ||||||
8 | (1,8) | . | . | . | (8,8) | ||
5 | . | . | . | . | . | ||
4 | . | . | . | . | . | ||
3 | . | . | . | . | . | ||
1 | (1,1) | (3,1) | (4,1) | (5,1) | (8,1) | ||
1 | 3 | 4 | 5 | 8 | B |
Statement 1 is true, while statement 2 is false.
We compared (1) and (2) with the similar statements:
(1) For any a there is a number b so
that a+b=a.
(A a) (E b)
(a+b=a)
(2) There is a number b so that
for any a a+b=a.
(E b) (A a)
(a+b=a)
In this case both statements are true, because we can use the number
0 for b.
Thinking about the examples we saw that in any case where there
is a switching of universal and existential quantifiers in their relative
position in the logical structure of a statement, the statement of form
2 will imple the statement of form 1.
Informally this means that if you have some b that works for
all a, you certainly can use that b to respond to the need to show the
existence of a b that would work for any specific a.
We next looked at determining the truth of compound statements and how this can be determined by the use of "truth tables" We constructed the basic truth tables for ~A, A -> B, A & B, A v B, and A<->B, and examined briefly an example of using these to determine a truth table for a mor complicated compound statement. Here's what a truth table for the basic connectives looks like:
A | B | ~A | A->B | A&B | AvB | A<->B | |
T | T | F | T | T | T | T | |
T | F | F | F | F | T | F | |
F | T | T | T | F | T | F | |
F | F | T | T | F | F | T |