Numbers and Sets

A set is a collection of objects. So, any collection of things, such as numbers, can be called a set. To show that we have a set, we use brackets { } To name a set we use capital letters.

A few others things to know about sets: means is an element of. means is NOT an element of. or { } means the empty set and contains no elements. For example: Given the set A = {1, 2, 3, 4, 5, 6, 7} we can say that but we would read this as “4 is an element of A”. we would read this as “9 is not an element of A”.

A subset: A set is a subset of another if every element in one set is also an element in the other set. To write this, we use the notation For example: Given P = {2, 3, 5} and Q = {1, 2, 3, 4, 5, 6}, then we can say that this is read “P is a subset of Q.”

The intersection of two sets, or groups is the elements that both groups have in common. Intersection: Here are two groups. What would the intersection of these groups be? A = {apples, oranges, pears}B = {apples, plums, pears} Answer: The intersection is {apples, pears} The word AND is used to mean an intersection. You can also use the symbol:

The union of two sets, or groups is the combination of the elements in both groups. It doesn’t matter if they are in common or not. Union: Here are two groups. What would the union of these groups be? A = {apples, oranges, pears}B = {apples, plums, pears} Answer: The union is {apples, oranges, pears, plums} The word OR is used to mean a union. You can also use the symbol:

For example: If P = {1, 3, 4} and Q = {2, 3, 5}, then: this is read “the union of P and Q is the set 1, 2, 3, 4, 5. this is read “the intersection of P and Q is the set 3.” If P = {3, 5, 7} and Q = {1, 2, 3, 4, 5}, then: this is read “the union of P and Q is the set 1, 2, 3, 4, 5, 7. this is read “the intersection of P and Q is the set 3 and 5.”

Given that M = {2, 3, 5, 7, 8, 9} and N = {3, 4, 6, 9, 10}: Are the following true or false? Answers: a)False b) True Given that M = {2, 3, 5, 7, 8, 9} and N = {3, 4, 6, 9, 10}: Give a list of the following. Answers: a){3, 9} b) {2, 3, 4, 5, 6, 7, 8, 9, 10} Given that M = {2, 3, 5, 7, 8, 9} and N = {3, 4, 6, 9, 10}: Are the following true or false? Answers: a)False b) True

Number Sets The set of all Counting Numbers is The set of all Natural Numbers is The set of all Integers is The set of all Positive Integers is The set of all Negative Integers is

Rational Numbers - Rational numbers can be written as the quotient of integers. Rational numbers can be written as decimals that either repeat or end. What does quotient mean? What do we mean when we say that decimals either repeat or end?

Irrational Numbers - Irrational numbers CANNOT be written as quotients of integers. Irrational numbers are decimals that continue forever but do NOT repeat. Can you write down an example of an irrational number? Some examples of irrational numbers are:

Real Numbers - Real numbers are ALL of the different types of numbers that you probably know right now. There are other types of numbers but we will learn about these later.

So, here is how these number systems all look like: Z: The Integers The Positive Integers Zero The Negative Integers The Real Numbers The Rational Numbers The Irrational Numbers

Our Number System Follows this Diagram: Real Numbers Rational Numbers Irrational Numbers Integers Natural Numbers