Use LEFT and RIGHT arrow keys to navigate between flashcards;
Use UP and DOWN arrow keys to flip the card;
H to show hint;
A reads text to speech;
29 Cards in this Set
- Front
- Back
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+x-2=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+x-2=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) x|x∈X or x∈Y |
|
The (1) consists of all elements belonging to both X and Y.
1. X∩Y= {(2)} |
(1) Intersection
(2) x|x∈X and x∈Y |
|
The (1) consists of all elements in X that are not in Y.
1. X-Y= {(2)} |
(1) Difference
(2) x|x∈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. A-B = (3) 4. B-A = (4) In general, A-B (5) B-A |
(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. Q-R = (3) |
(1) R
(2) Q (3) ∅ |
|
The set R-Q 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 U-X 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) X-bar or X^c (3) Universe |