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What are the two types of reasoning we use in everyday life?
- 1. Inductive reasoning
- 2. Deductive reasoning
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inductive reasoning
Inductive reasoning is the process of reasoning that arrives at a general conclusion or conjecture based on the observation of specific examples.
**Using specific examples to --- arrive --- at a general conclusion
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deductive reasoning
Deductive reasoning is the process of reasoning that arrives at a specific conclusion based on previously accepted general statements.
** Arrive at a specific conclusion --- via --- previously accepted statements.
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George Polya
FOUR major steps of problem solving
- 1. Understand the problem
- 2. Devise a plan
- 3. Carry out the plan
- 4. Look back, check the work
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estimation
Estimation is the process of finding an approximate answer to a mathematical problem.
In many cases, it is not necessary to find the exact answer to a problem. When only an approximate answer is needed, you can use estimation. This is accopmplished by rounding the numbers used in the problem, then performing the necessary operatioin or operations.
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define a set
A set is a well-defined collection of objects/elements.
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What is meant when a set is well defined?
A set is said to be well defined when there is no misunderstanding as to whether or not an element belongs to a set.
Ex:the set of "letters of the English alphabet" is a well-defined set since it consists of the 26 symbols.
- The set of Great Lakes is {Ontario, Erie, Huron, Michigan, Superior}
- The days of the week are {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}
- Write the set of natural numbers less than 6 {1, 2, 3, 4, 5}
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Name the 3 ways to designate sets
- 1. list or roster method
- 2. descriptive method
- 3. set-builder method
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Each object of a set is called...
an element or member of the set.
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Write the set of natural numbers less than 8.
{1, 2, 3, 4, 5, 6, 7}
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Write the descriptive of the set containing 2, 4, 6, 8, ...
Since the elements in the set are called the even natural numbers, the anser is:
E = even natural numbers
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Use set-builder notation to designate the set {2, 4, 6}
{ x | x ∈ E and x < 7 }
Reads: "The set of all x such that x is an even natural number and x is less than seven."
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Use set-builder notation to designate the set {red, yellow, blue}
{x | x is a primary color}
Reads: "The set of all x such that x is a primary color."
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Designate the set 32, 33, 34, 35, ... using the roster method.
{32, 33, 34, 35, ...}
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Designate the set 32, 33, 34, 35, ... using the descriptive method.
Natural numbers greater than 31.
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Designate the set 32, 33, 34, 35, ... using the set-builder notation.
{ x | x ∈ N and x > 31 }
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finite set
A set is said to be a finite set if the number of elements contained in the set is either 0 or a natural number.
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infinite set
A set is said to be an infinite set if it has an unlimited number of elements.
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finite? or infinte?
{the natural numbers that are multiples of 6}
infinite
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finite? or infinte?
{ x | x is a member of the U.S. Senate}
finite
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finite? or infinte?
{3, 6, 9, ..., 24}
finite
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null set
A set with no elements is called an empty set or null set. The symbols used to represent the null set are { } or circle w/line
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Two sets are equal if...
they have exactly the same members or elements.
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Two sets have a one-to-one correspondence if and only if...
it is possible to pair the elements of one set with the elements of the other set in such a way that for each element in the first set there exists one and only one element in the second set.
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What is a subset?
When all, some, or none of the elements of one set are used in another set, the second set is called a subset of the original set. Formally defined.
Ex: Subsets of {bacon, egg}
- {bacon, egg}
- {bacon}
- {egg}
- { }
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What is a proper subset?
If a subset of a given set is NOT equal to the original set, then the subset is called a proper subset of the original set.
Original set: {bacon, egg}
{bacon, egg} ⊄ {bacon, egg} **Because it is equal to the original
{bacon}⊂ {bacon, egg}
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What is the union of two sets?
All the elements of each set. Duplicates are not written twice.
Ex: A={10, 12, 14, 15} and B={13, 14, 15, 16, 17}
A ∪ B={10, 12, 13, 14, 15, 16, 17}
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What is the intersection of two sets?
The set of elements that are common to both sets.
Ex: A={10, 12, 14, 15} and B={13, 14, 15, 16, 17}
A∩ B = {14, 15}
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What is the complement of a set?
- It is the set of elements contained in the universal set that are NOT contained in the set noted.
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- A (the line over the set name denotes the compliment of that set)
- Basically everything outside of the mentioned set.
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What is a Venn Diagram?
When a set or sets are represented pictorially using Venn Diagrams.
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