Mat 243 Sets & Set Operations

What is the cardinality of the empty set?

0

What is the cardinality of the singleton set?

1

Enter T or F depending on whether the statement is true or false. (You must enter T or F -- True and False will not work.) 1. If A and B are disjoint sets, then c(A∪B)=c(A)+c(B) . 2. If A and B are not disjoint sets, then c(A∪B)<c(A)+c(B).

1. TRUE. If they are disjoint the their union A U B = A + B |A ∪ B| = |A| + |B| 2. TRUE because the formula for non disjoint sets is (A U B) - (A intersection B)- the elements in common. |A ∪ B| = |A| + |B| − |A ∩ B|.

What is an interval?

A set of all real numbers between two fixed numbers that includes or excludes each of these two numbers

Let A be the following set. A = {∅, 1, {1,2}}. Mark each of the following true T or false F. a. {{1,2}}⊆A b. {{1}}⊆P(A) c. {∅}∈P(A) d. {∅,1}∈P(A) e. (1,2)∈A×A

A. T B. T C. T D. T E. F

explain open and closed intervals

An open interval is an interval that does not contain its own end points: [𝑎,𝑏] = 𝑥∈R𝑎<𝑥<𝑏 A closed interval is an interval that contains its own end points: (𝑎,𝑏) = 𝑥∈R𝑎≤𝑥≤𝑏 There are also intervals which are half-open, half-closed: (𝑎,𝑏] = 𝑥∈R𝑎<𝑥≤𝑏 [𝑎,𝑏)= 𝑥∈R𝑎≤𝑥<𝑏

What type of elements can a set contain?

Any elements...does not have to be numbers and can even be other sets.

What is the complement?

By the complement of a set 𝑆, we mean all objects that are not contained in 𝑆. We write 𝑆ҧ for the complement of 𝑆. Some authors also use the notations 𝑆𝑐or 𝑆′. The complement is the set theoretic negation.

Describe Set equality

If a is a subset of B and B is a subset of A then they are said to be equal

Describe how elements of a set are distinguishable objects?

If you list the same element twice it does not mean that their are two but rather still one of them. An element can only be a member of a set or not a member. Not a double member. {1,1,1,1,1} = 1

what is the singleton set

It is the set that contains only one element. Ex a = {1} a = {{1,2}} C = {Ø}

Are cartesian products commutative?

No!

what is the difference between rational and irrational numbers.

Rational decimals repeat while irrational decimals dont repeat but continue indefinitely - ex pi 3.14....

What is the Difference of sets?

The difference of one set minus another set is the set of all elements that are in the first but not the s second set: 𝐴 − 𝐵 = 𝐴\𝐵 = {𝑥|𝑥 ∈ 𝐴 ∧ 𝑥 ∉ 𝐵} To the right is a Venn diagram illustration of 𝐴 − 𝐵. Unlike the previous set operations, the difference is not symmetric. 𝐴 − 𝐵 and 𝐵 − 𝐴 are not equal. There is no single logical operator that we named that corresponds to the difference operation. Example: if 𝐴 = 1,2,3 and 𝐵 = 3,4,5 , then 𝐴 − 𝐵 = 𝐴= 4,5. 1,2 and 𝐵 − Observe that we can use the difference to give an equivalent definition of the symmetric difference: 𝐴 △ 𝐵 = (𝐴 − 𝐵) ∪ (𝐵 − 𝐴).

What is the INTERSECTION of two sets?

The intersection of two sets is the set of all elements that are in both sets: 𝐴 ∩ 𝐵 = {𝑥|𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵} The intersection is the set theoretic equivalent of the and. To the right is a Venn diagram illustration of the intersection. Example: If𝐴= 1,2,3 and𝐵= 3,4,5 ,then𝐴∩𝐵={3}. If the intersection of 𝐴 and 𝐵 is empty, 𝐴 and 𝐵 are called disjoint. This situation is represented by a Venn diagram containing two circles that do not intersect. A collection of sets is called pairwise disjoint if any two of them are disjoint.

Describe how a set is an unordered data structure?

The order of the elements inside the set don't matter.

Explain the Universal Set.

The set of "all" objects that the complement depends on is called the universal set, and usually denoted 𝑈. With that, we can give a formal definition of complement in terms of a set difference: 𝐴ҧ = 𝑈 − 𝐴

What is the cartesian product?

The set of all ordered pairs (𝑎, 𝑏), where 𝑎 is in some set 𝐴, and 𝑏 is in some set 𝐵 is called the Cartesian product of 𝐴 and 𝐵: All ordered pair you man manufacture from elements a and b. For example, if 𝐴 = {1,2} and 𝐵 = {3,4} then 𝐴×𝐵={1,3, 1,4, 2,3, 2,4} to check to see if you have enough pairs just look at cardinality - rule is |A x B| = A times B - how many ordered pairs you should have.

What is the difference between rational and real numbers.

The set of real numbers are is the set of rational and irrational numbers together

What is the UNION of two sets?

The union of two sets is the set of all elements that are in one or the other set: 𝐴 ∪ 𝐵 = {𝑥|𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵} The union is the set theoretic equivalent of the inclusive or. To the right is a Venn diagram illustration of the union. Example: If 𝐴 = 1,2,3 and 𝐵 = 3,4,5 , then 𝐴 ∪ 𝐵 = {1,2,3,4,5}.

If A is a proper subset of B, then A is a subset of B.

True

The intersection of the empty set and the universal set is the empty set.

True

How do we denote a is a member of a set?How do we negate it.

a ∈ A negation: a ∉ A

What is a set?

an unordered collection of distinguished objects. Thes objects are called members or elements of a set. Sets are usually denoted with uppercase letters, and their members by lowercase letters.

To define a finite set we puts its elements in...

braces A = {1,2,3}

What is the SYMMETRIC DIFFERENCE of two sets?

he symmetric difference of two sets is the set of all elements that are contained in exactly one of the sets: 𝐴 △ 𝐵 = 𝐴⨁𝐵 = {𝑥|(𝑥 ∈ 𝐴)⨁(𝑥 ∈ 𝐵)} The symmetric difference is the set theoretic equivalent of the exclusive or. To the right is a Venn diagram illustration of the symmetric difference . Example: If 𝐴 = 1,2,3 and 𝐵 = 3,4,5 , then 𝐴 △ 𝐵 = {1,2,4,5}.

Describe a subset relationship.

if A is a subset of B that means that every element of set A is also in set B.

What is a proper subset?

if set a is a subset of b and b contains an element that is not in a then a is a proper subset of b.] {1,2} ⊆ {1,2,3} and {1,2} ⊂ {1,2,3}. Every set is a subset of itself but not a proper subset of itself

What is an ordered pair?

is an ordered list of two objects: (𝑎, 𝑏). We can think of an ordered pair of two numbers as a point. (Here, the notation (𝑎, 𝑏) does not refer to an open interval.)

What is the empty set

it is a set that contains no elements

Use set builder notation to say that s = the set containing x, which is an integer between 1 and 100 inclusive.

s = {x|x is an integer and 1≤x≤100} same thing as s = {1,2,3,4....100} but only use first notation to reduce ambiguity.

C symbol

set of complex numbers {z | z=a+bi, -∞<a<∞, -∞<b<∞} 6+2i

double z symbol

set of integers -3,-2,-1,0,1,2,3

double N symbol

set of natural numbers(whole numbers) 1,2,3,4,5 or N sub 0 including zero

Q symbol

set of rational numbers 1/2,1/3...

R symbol

set of real numbers {x | -∞ < x <∞} 6.345...

What is cardinality?

the number of elements in a set

what is a power set?

the set of all subsets of a set

Let W={a,b,c}, X={1,2,3} Determine X−W

{1,2,3}

Let A=ℝ, B=ℕ Determine A∩B

ℕ

Commutitive Law of Intersection

𝐴∩𝐵= 𝑥𝑥∈𝐴∧𝑥∈𝐵 = 𝑥𝑥∈𝐵∧𝑥∈𝐴 =𝐵∩𝐴

𝒫 {1} = 𝒫 {1,2} =

𝒫 {1} = ∅, 1 𝒫 {1,2} = ∅, 1 , 2 , 1,2 .