This is not a simple 1-2-3-4 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.
One comment about the homeworks: Most people answer only part of any
given problem.
Determine what you have and what you want.
To indicate an answer clearly to someone else (like me), you need to know
what the answer is.
Consider rephrasing the problem, either in English or symbolically.
Rephrasing the problem may help you remember solution methods.
Try some examples.
This is close to devising a plan. Sometimes you may stumble upon an
answer.
Look for relationships between the data.
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:
Guessing and checking.
Try a few combinations of the data. See what falls out. This is good for
finding relationships and understanding the problem.
Searching using a list.
If you know the answer lies in some range, you can search that range
systematically by building a list.
Finding patterns.
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.
Following dependencies / working backwards.
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
sub-problems.
When building a list, be sure to carry out a well-defined procedure. Or when looking
for patterns, be systematic in the examples you try. Don’t jump around
randomly.
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\}.
empty set
The unique set with no elements: \{\}
or ∅.
Be sure to know the relations between the empty set and other sets, and
also how the empty set behaves in operations.
element of
You write x A
to state that x
is an element of A.
The symbol is not an “E” but is almost a Greek ϵ.
Think of a pitchfork.
Note that 1 \{1,2\}
and \{1\} \{\{1\},\{2\}\},
but \{1\}∉\{1,2\}.
subset
Given two sets A
and B,
A ⊂ B
if every element of A
is also an element of B.
One implication is that ∅⊂ A
for all sets A.
This statement is vacuously true.
Here \{1\} ⊂\{1,2\}
and \{1\}⊄\{\{1\},\{2\}\}.
superset
Given two sets A
and B,
B ⊃ A
if every element of A
is also an element of B.
proper subset or superset
A subset or superset relation is proper if it
implies the sets are not equal.
Venn diagram
A blobby diagram useful for illustrating operations and
relations between two or three sets.
union
A ∪ B = \{x\kern 1.66702pt |\kern 1.66702pt x A\text{ or }x B\}.
The union contains all elements of both sets.
intersection
A ∩ B = \{x\kern 1.66702pt |\kern 1.66702pt x A\text{ and }x B\}.
The intersection contains only those elements that exist in both sets.
set difference
A \ B = \{x\kern 1.66702pt |\kern 1.66702pt x A\text{ and }x\mathrel{∉}B\}.
The set difference contains elements of the first set that are not in the
second set. It cannot contain any elements of the second set.
You can write the result of multiple operations in set-builder notation,
\eqalignno{
(A ∩ B) ∪ C & = \{x\kern 1.66702pt |\kern 1.66702pt x A\text{ and }x B\} ∪ C & &
\cr
& = \{x\kern 1.66702pt |\kern 1.66702pt (x A\text{ and }x B)\text{ or }x C\}. & &
}
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 one-to-one 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 one-to-one correspondence is
straight-forward. 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 straight-forward 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.
For any operation, the result of the operation always lies in the set. For
whole numbers, x + y
is closed but x - y
is not.
commutative
For numbers, x + y = y + x
and xy = yx.
For sets, A ∩ B = B ∩ A
and A ∪ B = B ∪ A.
associative
For numbers, x + (y + z) = (x + y) + z
and x(yz) = (xy)z.
For sets, A ∩ (B ∩ C) = (A ∩ B) ∩ C)
and A ∪ (B ∪ C) = (A ∪ B) ∪ C.
distributive
For numbers, the only version is the distribution of multiplication
over addition or x(y + z) = xy + xz.
For sets, both unions and intersections distribute. So A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
and A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C).
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| + |B|-|A ∩ B|-|A| + |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.
A
to state that x
is an element of A.
The symbol is not an “E” but is almost a Greek ϵ.
Think of a pitchfork.