2.1 Symbols and Terminology

Designating Sets A set is a collection of objects (in math, usually numbers). The objects belonging to the set are called the elements (or members) of the set. – A set containing no elements is called the empty set (or null set) symbolized as Ø or { }, but NOT {Ø}! Sets are designated using the following three methods: 1.Word Description “The set of even counting numbers less than 10.” 2.Listing Method {2, 4, 6, 8} 3.Set-builder Notation {x|x is an even counting number less than 10}

Sets of Numbers Positive Integers (aka Natural/Counting Numbers): {1, 2, 3, 4, …} Natural Numbers (aka Whole Numbers): {0, 1, 2, 3, 4, …} Integers: {…, -3, -2, -1, 0, 1, 2, 3, …} Rational Numbers: { p / q where p and q are integers, q ≠ 0} – Includes all integers and any fraction that can be expressed as either a terminating or repeating decimal. Real Numbers: any number that can be written as a decimal Irrational Numbers: a real number that CANNOT be written as a quotient of integers – Examples include √2, π, and all non-terminating and non- repeating decimals.

Listing Elements of Sets  Give a complete listing of all the elements of each of the following sets. – The set of counting numbers between 6 and 13 – {5, 6, 7, …, 13} – {x|x is a counting number between 6 and 7}

Applying the Symbol  When something belongs to a set, we use the symbol  to say that it “is an element of” the set. Similarly, we use  when it “is not an element of” the set.  Decide whether each statement is true or false. – 3  {1, 2, 5, 9, 13} – 0  {0, 1, 2, 3} – 

Cardinality The number of elements in a set is called the cardinal number or cardinality of the set. The symbol n(A), which is read “n of A”, represents the cardinal number of set A. – If elements are repeated in a set listing, they should only be counted once! – For example: B = {1, 1, 2, 2, 3} and n(B) = 3

Finding Cardinal Numbers  Find the cardinal number of each set. K = {2, 4, 8, 16} M = {0} R = {4, 5, …, 12, 13} Ø

Finite and Infinite Sets If the cardinal number of a set is whole number (0 or a positive integer), we call that set a finite set. – There is only a certain number of elements and (given enough time) all elements could be counted. Whenever a set is so large that its cardinal number cannot be found, we call that set an infinite set. – No matter how long we try, we cannot count all the elements (for example, the set of integers is infinite).

Designating an Infinite Set Designate all odd counting numbers by the three common methods of set notation. 1.Word description – The set of all odd counting numbers  2. Listing method  3. Set-builder notation

Equality of Sets Set A is equal to set B provided the following two conditions are met: 1.EVERY element of A is an element of B 2.EVERY element of B is an element of A For example: – {a, b, c, d} = {a, c, d, b} – {1, 0, 1, 2, 3, 3} = {0, 1, 2, 3} – {-3, -2, -1, 0} ≠ {-2, -2, 0, 1}

Determining Whether Two Sets are Equal  Decide whether each statement is true or false. {-4, 3, 2, 5} = {-4, 0, 3, 2, 5} {3} = {x|x is a counting number between 1 and 5} {x|x is a negative natural number} = {y|y is a number that is both rational and irrational}