Contemporary Mathematics Topic 1 Intro to Sets What

Contemporary Mathematics Topic 1: Intro to Sets

What Is A Set? ØDefine the word “set” in your own words. ØA set is a familiar word in everyday life. It also has an importance in mathematics. ØA set is a well-defined collection of objects. These objects are called elements. ØThe main property of a set in mathematics is that it is well-defined. What does this mean? ØThe set of odd whole numbers is {1, 3, 5, 7, 9, …}. This is well-defined. The set of “lucky numbers” is not.

Example 1: › Which of the following descriptions define a welldefined set? › A) Funny movies › B) Current President of the United States › C) Powers of 5 › D) Outcomes of rolling a 4 -sided die › E) Numbers that can be substituted for x such that x+6 = 10

Elements of a Set ›

Set-Builder Notation › In set-builder notation, the set is described with a variable, a vertical bar | (read as “such that”), and a description of the rule for that set. › If we have a set of all natural numbers less than 8, the way to write that in set-builder notation is {x | x is a whole number less than 8} › The variable x just represents any element of the set. › Another example: If we want all counting numbers less than 15, we would write {x | x is a counting number less than 15}

Example 2: › Describe the following in list form (also called roster), and then in set-builder notation. › A) The set of digits in 1492 › B) The set of odd whole numbers greater than 11 › C) The set of whole number multiples of 4 › D) The set of vowels in the English alphabet › E) The odd digits of the current year

Empty Set ›

Special Sets ›

Equality of Sets › Two sets are considered equal if they have the same elements. › Order does not matter. › Repeated elements do not matter. › Example: {3, 1} = {1, 3, 3, 3}

Subsets of a Set ›

Subsets (continued…) ›

Example 3: › A) List all subsets of A = {a, b, c} › B) How many subsets will B = {1, 2, 3, 4, 5, 6, 7, 8} have? › C) How many proper subsets will C = {a, b, c, … , z} have? (A formula for this suffices – no need to actually list the number!)