This section will discuss the symbolism and concepts of set theory 6.2 Sets This section will discuss the symbolism and concepts of set theory

Set properties and set notation Definition of set: A set is any collection of objects specified in such a way that we can determine whether or not an object is or is not in the collection. Example 1. Set A is the set of all the letters in the alphabet. Notation: A = { a, b, c, d, e, …z) We use capital letters to represent sets. We list the elements of the set within braces. The three dots … indicate that the pattern continues. We can determine that an object is or is not in the collection. For example . e A stands for “e is an element of , or e belongs to set A” . This statement is true. The statement “ 3 A is false, since the number 3 is not an element of set A. The statement “ 3 ∈ A” is true.

Null set Example. What are the real number solutions of the equation? Answer: There are no real number solutions of this equation since no real number squared added to one can ever equal 0. We represent the solution as the null set { } or Ø

Set builder notation Sometimes it is convenient to represent sets using what is called set-builder notation. For example, instead of representing the set A, letters in the alphabet by the roster method, we can use set builder notation: means the same as { a , b , c, d, e , …z} Example two. { x l } = {3 , -3} . This is read as the set of all such that the square of x equals 9. The solution set consists of the two numbers 3 and -3.

Subsets A B means A is a subset of B. A is a subset of A if every element of A is also contained in B. For example, the set of integers denoted by { …-3, -2, -1, 0, 1, 2, 3, …} is a subset of the set of real numbers. Formal definition of subset: A B means if x A, the x B (null set )is a subset of every set. To verify this statement, let’s use the definition of subset. “ if x , then x is an element of A. But since the null set contains no elements, the statement x is an element of the null set is false. Hence, we have a conditional statement in which the premise is false. We know that p q is true if p is false. Since p is false, we conclude that the conditional statement is true. That is “ if x belongs to the null set, then x belongs to set A” is true, which implies that the null set must be a member of every set. Therefore, the null set is a subset of every set.

Number of subsets List all the subsets of set A = {bird, cat, dog} For convenience, we will use the notation A = {b , c, d} to represent set A. Solution: is a subset of A. We also know that every set is a subset of itself so A = {b , c, d } is a subset of set A since every element of set A is contained within set A. How many two-element subsets are there? We have {b, c}, {b, d} , {c, d} How many one-element subsets? { b} , {c} and {d} . There is a total of 8 subsets of set A if you count all the listed subsets.

Set operations The union of two sets is the set of all elements formed by combining all the elements of set A and all the elements of set B into one set. The symbolism used is The Venn Diagram representing the union of A and B is the entire region shaded yellow. A B

Example of Union The union of the rational numbers with the set of irrational numbers is the set of real numbers. Rational numbers are those numbers that can be expressed as fractions, while irrational numbers are numbers that cannot be represented exactly as fractions, such as Rational numbers a/b, ¾, 2/3 , 0.6 Irrational numbers such as square root of two, Pi , square root of 3 Real numbers: represented by shaded blue-green region

Intersection of sets A and B The intersection of sets A and B is the set of elements that is common to both sets A and B. It is symbolized as { x l x ∈A and x ∈B } Represented by Venn Diagrams: B A Intersection

Complement of a set A’ = { x ∈ U |x∈ A} To understand the complement of a set, we must first define the universal set. The set of all elements under consideration is called the universal set. For example, when discussing numbers, the universal set may consist of the set of real numbers. All other types of numbers (integers, rational numbers, irrational numbers ) are subsets of the universal set of real numbers. Complement of set A: The complement of a set A is defined as the set of elements that are contained in U, the universal set, but not contained in set A. The symbolism for the complement of set A follows: A’ = { x ∈ U |x∈ A}

Venn Diagram for complement of set A The complement of set A is represented by the regions that are colored blue and yellow. The complement of set A is the region outside of the white circle representing set A. Yellow region= all elements in U that are neither in A or B. A Elements of set B that are not in A A’ B A’