Download §2.1: Basic Set Concepts MGF 1106-Peace Def: A set is a collection

§2.1: Basic Set Concepts
MGF 1106-Peace
Def: A set is a collection of objects that can be clearly determined.
Def: A collection of objects that is not well defined cannot be a set.
Def: The objects within a set are called members or more commonly elements.
We use sets in our daily lives. For example, when you wash your clothes, you (hopefully) divide your laundry into
different sets based on color or fabric type. Where else do you use sets?
I.
Representing Sets
Generally sets are named using capital letters.
a. Word Description
Let the set M be the set of all the months in a year. We described the set implicitly with
words.
b. Roster Method-In this method we actually list out all the elements in the set.
M = {January, February, March, April, …, October, November, December}
-Always use braces when representing sets this way (not brackets or parentheses)
-Order is not important when listing the elements of a set.
c.
Set Builder Notation
-This is probably the most elegant way to represent sets. Here is the form:
-M = {x| x is a month of the year}
-Spoken as “The set M is the set of all elements x such that x is a month of the year.”
-“x” is a dummy variable and can be any variable we want.
-The vertical bar represents the phrase “such that” in math.
You should be able to represent sets in all three ways.
Ex. Determine if the following collection is well defined and therefore a set:
“The collection of elderly full time students currently enrolled at LCCC.”
**This collection is not well defined since different people would argue about with “elderly” means. Some would say
60 is elderly, others 70, some might even say 50. Because this is not well defined, it is NOT a set**
II.
The Empty Set
Believe it or not, we need to have a set that contains no elements. It is STILL a set, it just doesn’t have anything in it.
We call this type of a set the empty set. It is still a set because it is well defined, but no elements belong to it.
Representations for empty set: { } (empty braces) or φ (Greek Letter phi)
Ex. Let W = {x | x is a female President of the US elected prior to 2000}.
Since there have been no female US presidents elected prior to 2000, this is the empty set: { } or φ
Ex. Is {0, φ} the empty set?
Ex. Is 0 the empty set?
This set is a set containing 2 elements,
0 isn’t even a set, so it can’t be the empty set.
So it is NOT the empty set.
III.
Element Notation
The notation ∈ is used to indicate that an object is an ELEMENT of a set.
The notation ∉ is used to indicate that an object is NOT an ELEMENT of a set.
Here’s how it works: “object” ∈ or ∉ “Set”
Ex. If A = {2, 4, 6}, then 2 ∈ A and -2 ∉ A
Ex. Fill in the blank: If B = the set of all multiples of 4, then 54 ______ B
IV.
Sets of Natural Numbers
-Recall the natural numbers are all whole numbers starting at 1 and going up: {1, 2, 3, 4, …}
-Since it comes up so much, we have a special name for this set: N = the set of all natural numbers.
This is an infinite set, meaning it goes on forever.
Ex. Represent the following set in roster method: The set of all natural numbers less than 6.
This would be the natural numbers from 1-5: {1, 2, 3, 4, 5}
Ex. Represent the following set in roster method: {x | x ε N and 15 < x < 20}. Hint: the “and” means both conditions
must be met.
List all natural numbers greater than 15 but also less than 20: {16, 17, 18, 19}
V.
Cardinality
The number of distinct elements in a set is called the cardinal number or cardinality of the set.
Notation: n(A) represents the cardinality of the set A.
Ex. A = {2, 6, 8} has a cardinal number of 3 since there are three distinct elements, so n(A) = 3.
Note: Any repeated elements are not counted again for cardinal number purposes.
Ex. Calculate the cardinal number of B = {x | x ε N and 3 < x < 10}.
Write the set is roster method then count of the number of elements to get the cardinality.
B = {4, 5, 6, 7, 8, 9}. There are 6 elements in B so n(B) = 6.
Ex. Calculate the cardinal number of C = {y | y < 5 and y > 15}
VI.
Equivalent and Equal Sets
Def: Two sets A and B are said to be equivalent if their cardinal number is the same: n(A) = n(B).
Def: Two sets A and B are said to be equal if the sets contain EXACTLY the same elements regardless of order and
repetition. Notation if two sets A and B are equal, we say A = B.
Ex. Determine if the two sets are equivalent and then if they are equal:
A = {3, 2, 1, 0}, B = {0, 0, 1, 1, 2, 3, 2, 3, 3, 0, 1, 2}
Remember two sets are equivalent if they have the same cardinality. Remember also that cardinality only counts
DISTINCT elements (don’t count repeats). So n(A) = 4 and n(B) = 4. Therefore A and B are equivalent. Since A
and B contain the same 4 elements {0, 1, 2, 3} they are also equal: A = B.
Ex. Determine if the two sets are equivalent and then if they are equal:
A = {x | x ε N and 12 < x < 17}, B = {x | x ε N and 20 < x < 25}
Notice if A and B are equal, they MUST be equivalent. But if they are equivalent, they may or may not be equal.
VII.
Finite and Infinite Sets
-Set A is a finite set if it has cardinality of 0 (the empty set) or has cardinality that is natural number. Set A is infinite if it
has cardinality that is not 0 or is not a natural number.
-As stated before the set N (the natural numbers) is an infinite set. Strangely enough we do not say the cardinality is ∞.
Instead we give it the cardinality of ‫א‬0 which is the smallest of the infinities. Yes, there are different “levels” of infinity.
This one is at the lowest levels. Other levels of infinity are ‫א‬1, ‫ א‬2, etc. which represent higher levels of infinity.
Ex. Finite or Infinite? {x | x ε N and x > 1,000,000}
This set is infinite because it goes on forever: {1,000,001, 1,000,002, …}
Ex. Finite or Infinite? {x | x ε N and x < 5,000,000}
Applications (Working Women and Happiness throughout the day)
Represent the following set in roster method: {x | x is a time of day when 3 < average level of happiness < 4}
You try this one.