Algebra Textbook: Book 1 - Set Theory/Basic Set Concepts

Basic Set Concepts Definition of a Set A set is a well-defined collection of distinct objects. These objects are called the elements or members of the set. Notation We typically denote sets using capital letters (A, B, X, Y, etc.) and elements using lowercase letters (a, b, x, y, etc.). If an element x belongs to a set A, we write: If an element x does not belong to a set A, we write: Ways to Describe Sets There are several ways to describe or specify sets: Roster Notation (Listing Method): List all elements of the set within curly braces. Set-Builder Notation: Describe the set by stating a property that all elements must satisfy. This reads as "B is the set of all elements x in the integers such that x is greater than 0 and less than 6." Special Sets Several sets are so important they have their own symbols: or : The empty set (or null set) - a set containing no elements : The set of all natural numbers : The set of all integers : The set of all rational numbers (fractions) : The set of all real numbers : The set of all complex numbers Set Equality Two sets A and B are equal if and only if they contain exactly the same elements. This is denoted as . For example, because the order of elements doesn't matter in a set. Subsets A set A is a subset of set B if every element of A is also an element of B. This is denoted as . If A is a subset of B, but A ≠ B (meaning B contains at least one element that A does not), then A is a proper subset of B, denoted as . Properties of Subsets Every set is a subset of itself: The empty set is a subset of every set: If and , then (transitivity) Power Set The power set of a set A, denoted or , is the set of all possible subsets of A, including the empty set and A itself. For example, if , then: For a finite set with n elements, its power set has elements. Cardinality of a Set The cardinality of a set A, denoted |A| or #A, is the number of elements in the set. For example: Universal Set In a particular context or discussion, the universal set, often denoted by U, is the set that contains all objects under consideration. Venn Diagrams Venn diagrams are graphical representations of sets as circles or other closed curves within a rectangle (representing the universal set). They are useful for visualizing set relationships and operations. Exercises 1. Determine whether each statement is true or false: a) b) c) d) e) 2.Write the following sets using roster notation: a) b) c) 3. Write the following sets using set-builder notation: a) b) c) 4. Find the power set of: a) b) c) 5. If , , and , determine: a) Which sets are subsets of other sets? b) The cardinality of each set c) Is ? Explain. d) Is ? Explain. 6. Draw Venn diagrams to represent: a) b) Sets A and B are disjoint c) Sets A and B have some elements in common, but neither is a subset of the other