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29 Cards in this Set
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A set is simply a collection of (1)

(1) Objects


Objects are sometimes referred to as (1) or (2)

(1) Elements
(2) Members 

Set numbers that occur frequently:
Z  (1) Q  (2) R  (3) 
(1) Zahlen (Integers)
(2) Quotient (Fractions) (3) Real Numbers 

Use superscript negative to denote (1)

(1) Negative Numbers that belong to said set


The cardinality is the number of (1) of x. It is denoted as (2)

(1) Elements
(2) x 

Example 1.1.1
The cardinality of the set {R,Z} is (1) since it contains (2), namely (3) and (4) 
(1) 2
(2) 2 Elements (3) R (4) Z 

The set with no elements is called the (1) and is denoted as (2); thus (2) = { }

(1) Empty Set
(2) ∅ 

Example 1.1.2
Two sets are equal when: 1. For every x, if x∈X, then (1) 2. For every x, if x∈Y, then (2) 
(1) y∈Y
(2) x∈X 

A= {1,3,2} and B= {2,3,2,1} are equal because (1)

(1) A and B have the same elements


Example 1.1.4
If every element of x is an element of y, we say that x is a (1) of y and write (2) 
(1) Subset
(2) x⊆y 

Example 1.1.5
If C= {1,3} and A= {1,2,3,4}, (1) because ever element of C is an element of A 
(1) C⊆A


Example 1.1.6
Let X= {x x²+x2=0}. Show that X⊆ℤ 1. First, show that (1) 2. If (1) is true, then (2), so (3) 3. Solve for x, conclude that, Since x∈ℤ, (4), or (5) 
(1) For every x, if x∈X, then x∈ℤ
(2) x∈ℤ (3) x²+x2=0 (4) X is a subset of ℤ (5) X⊆ℤ 

Example 1.1.7
The set of integers ℤ is a (1) of the set of rational numbers Q. If n∈ℤ, n can be expressed as (2). Therefore, n∈Q and ℤ(3)Q 
(1) Subset
(2) a quotient of integers (n=n/1) (3) ℤ⊆Q 

Example 1.1.8
The set of rational numbers Q is a (1) of the set of real numbers R. If x∈Q, x corresponds to (1); therefore, (2) 
(1) A point on the number line
(2) x∈R 

For x to not be a subset of y, there must be at least...(1)

(1) at least ONE member of x that is not in y


If x is a subset of y and x does not equal y, we say that x is a (1) of (2) and write (3)

(1) Proper Subset
(2) y (3) x⊂y 

Let C= {1,3} and A= {1,2,3,4}.
1. C is a (1) of A since C is a subset of A but C does not equal A; therefore (2) 
(1) Proper Subset
(2) C⊂A 

The set of (2) (proper or not) of set x, denoted P(X), is called the power set of X

(1) All Subsets


Example 1.1.13
If A= {a,b,c}, the members of P(A) are: ∅, {a}, {b}, {c}, {a,b}, {a,c}, {b,c}, {a,b,c} 1. All but (1) are proper subsets of A 2. A = (2) 3. P(A) = (3) 
(1) {a,b,c}
(2) 3 (3) 2^n elements = 8 

The (1) consists of elements belonging to either x or y or both
1. X∪Y= {(2)} 
(1) Union
(2) xx∈X or x∈Y 

The (1) consists of all elements belonging to both X and Y.
1. X∩Y= {(2)} 
(1) Intersection
(2) xx∈X and x∈Y 

The (1) consists of all elements in X that are not in Y.
1. XY= {(2)} 
(1) Difference
(2) xx∈X and x∈Y 

Example 1.1.14
If A= {1,3,5} and B= {4,5,6}, then: 1. A∪B = (1) 2. A∩B= (2) 3. AB = (3) 4. BA = (4) In general, AB (5) BA 
(1) {1,3,4,5,6}
(2) {5} (3) {1,3} (4) {4,6} (5) ≠ 

Example 1.1.15
Since Q⊆R 1. R∪Q = (1) 2. R∩Q = (2) 3. QR = (3) 
(1) R
(2) Q (3) ∅ 

The set RQ is called the set of (1), consists of all real numbers that are not (2)

(1) Irrational Numbers
(2) Rational 

The sets X or Y are (1) if X∩Y=∅. A collection of sets S is said to be (2) if whenever X and Y are distinct sets in S, X and Y are disjoint

(1) Disjoint
(2) Pairwise Disjoint 

A (1) is a set which contains all objects, including itself

(1) Universal Set


Example 1.1.16
The sets {1,4,5} and {2,6} are disjoint. The collection of sets: S= {{1,4,5}, {2,6}, {3}, {7,8}} are (1) 
(1) Pairwise Disjoint


Given a universal set U and a subset X of U, the set UX is called (1) and is written (2). The complement obviously depends on the (3) in which we are working

(1) The complement of X
(2) Xbar or X^c (3) Universe 