2.1 Unit 2: Sets and Functions

The material in this unit comprises 10% of the high-stakes assessment and includes such concepts as:

• Sets, subsets, and power sets
• Set operations
• Cartesian product of sets
• Set partitions
• Overview of functions
• Function properties

2.2 Module 6: Working with Sets

This page marks the beginning of Module 6. This module contains ten lessons:

• Set theory and terms
• Universal sets and subsets
• Set of sets
• Power sets
• Set operations
• Sets with multiple operations
• Set difference and symmetric difference
• Combining sets
• Set identity and laws
• Applying set identity laws

MODULE OBJECTIVE: By the end of this module, you should be able to analyze sets and set operations, including subsets, power sets, set cardinality, and set identities.

2.3 Lesson: Set theory and terms

• A set is a collection of objects. Objects may be of various types, such as titles of books, names of bridges, or rational numbers.
• The objects in a set are called elements. 
• The symbol ∈ is used to indicate that an element is in a set, as in 2 ∈ A
• The symbol ∉ indicates that an element is not in a set, as in 5 ∉ A
• The set with no elements is called the empty set and is denoted by the symbol ∅
• The empty set is sometimes referred to as the null set and can also be denoted by {}
• A finite set has a finite number of elements.
• An infinite set has an infinite number of elements.
• The cardinality of a finite set A, denoted by |A|, is the number of elements in A

• For example, the set R⁺ is the set of all positive real numbers, and Z⁺ is the set of all positive integers. 
• A number x is positive if x > 0.
• For example, the set R^– is the set of all negative real numbers, and Z^– is the set of all negative integers.
• A number x is negative if x < 0. The number 0 is neither positive nor negative, so 0 ∉ Z⁺ and 0 ∉ Z^–.
• A number x is non-negative if x ≥ 0
• In set builder notation, a set is defined by specifying that the set includes all elements in a larger set that also satisfy certain conditions
• The colon symbol “:” is read “such that”. 
• A = { x ∈ S : P(x) }
• The description for A above would read: “all x in S such that P(x)”.

Now that you have completed this lesson you should be able to find the set membership and cardinality from a set expressed in set builder notation. Take a moment to think about what you’ve learned in this lesson:

• A set is a collection of objects. The objects of a set are called elements. If x is an element in a set A, it is denoted x∈A
• A set can contain any type of object. It can contain a finite number of elements or infinitely many.
• A set that contains no elements is called the empty set or the null set and is denoted ∅
• Set builder notation is a way to represent a set. For example, the set of all real numbers bigger than 3 can be expressed as {x∈R|x>3}
• When describing a set of elements, it is expressed in context to a superset called the “universal set.” For example, {x∈R|x≥3} refers to the universal set of all real numbers. The bar notation | means “such that”, so {x∈R|x≥3} refers all real numbers greater than or equal to 3.

2.4 Lesson: Universal sets and subsets

• The universal set, usually denoted by the variable U, is a set that contains all elements mentioned in a particular context.
• Sets are often represented pictorially with Venn diagrams.
• If every element in A is also an element of B, then A is a subset of B, denoted as A ⊆ B If there is an element of A that is not an element of B, then A is not a subset of B, denoted as A ⊈ B. 
• If A ⊆ B and there is an element of B that is not an element of A (i.e., A ≠ B), then A is a proper subset of B, denoted as A ⊂ B.Venn diagrams are particularly useful for visualizing subset relationships between sets.

2.5 Lesson: Set of sets

Lesson summary

Now that you have completed this lesson you should be able to analyze sets of sets for membership. Take a moment to think about what you’ve learned in this lesson:

• The following is the answer to the question posed in the introduction: If S is in S then it cannot be a member of S by definition of being in S.
• Sets can contain sets as elements.
• The cardinality of a set within a set is 1. That is, each set counts as a single element.