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Linear Equations IV


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<-----○ (2)
x < 2
<-----● (3)
x ≤ 3
(-4) ●----->
x ≥ -4
<-----● (-4)
x ≤ -4
<-----○ (4/3)
x < 4/3
<-----○ (-2)
x < -2
(-6/5) ○----->
x > -1 1/5
<-----● (-9)
x ≤ -9
(1 1/4) ○----->
x > 1 1/4
Set
A _ is a well-defined collection of objects.
Natural numbers
A group of numbers that include 1,2, 3,.....
X = {5, 6, 7, 8, 9}
Set Y is the set of all natural numbers between 2 and 7, inclusive.
Y = {2, 3, 4, 5, 6, 7} is a finite set
If the elements in a set can be counted, the set is called a finite set. Otherwise, the set is infinite.
Another way to write a set is using set-builder notation.
This is useful when the individual elements are not easily written.
Set A is the set of all natural numbers greater than 4
A = {x | x is a natural number greater than 4}
A = {a, e, i, o, u}
A = {x | x is a vowel in the alphabet}
Set B is the set of natural numbers between -6 and 0.
Set B is an empty set, or a null set. We can write this as { } or Ø.
The notation { Ø } means "a set whose only element is the empty set."
When writing the notation for an empty set, only use one of the symbols -- either { } or Ø, but not both.
There are several ways to write a set:
C = {1, 2, 3, 4, 5, 6}
Intersections and Unions
Two operations that are used with sets are the intersection and the union.
Set Intersection
The intersection of two sets is the set of all the elements that are common to both sets.
Set Union
The union of two sets is the set of every element that is in either or both sets.
Intersection = C⋂D = {3, 9}
D = {3, 6, 9, 12, 15}
Subsets
When all of the elements of one set are contained in another set, the smaller set is a subset of the larger set.
B = {letters of the alphabet}
Set A is a subset of B. this can be written as A⊆B.
If A = {odd numbers} and B = {integers}, then A⊆B
True... A = {1, 3, 5, 7...} and B = {...-2, -1, 0, 1, 2, 3, ...}
If A = {multiples of 3} and B = {multiples of 4}, then A⊆B
False... A = {3, 6, 9, 12...} and B = {4, 8, 12, 16...}
The solution set is {0, 1, 2}
{x | x is a whole number less than 3}
The solution set is Ø
{x | x is a whole number less than 0}
A⋂B = The solution set is Ø
A = {3, 5, 6, 8, 9} and B = {0, 1, 2, 4}
A⋂B = {6, 7, 8, 9}
A = {2, 6, 7, 8, 9} and B = {6, 7, 8, 9}
True
If B = {2, 4, 6} and C = {1, 2, 3, 4, 5, 6, 7}, then B⊆C.
False
8 ϵ {x | x is an odd number}
{x | x is an integer number between -3 and 5}
The solution set is {-2, -1, 0, 1, 2, 3, 4}
B⋂C = The solution is Ø
B = {2, 3, 6, 9} and C = {1, 4, 5, 7, 8}
G = {4, 6, 8, 10}
G = {x | x is even and 4 ≤ x ≤ 10}
True
{16, 17, 18, 19} ⊆ {15, 16, 17, 18, 19}
Interval Notation is another way to represent the solution to an inequality.
It uses two values -- the starting point and the end point of the interval representing the solution. These two values are written inside (parentheses) and/or [brackets].
Infinity - an infinite set is a set whose elements cannot be counted.
An example of this is the set of real numbers. The set of real numbers increases without bound to the right and decreases without bound to the left on a number line.
Positive Infinity - the ∞ symbol is used to denote positive infinity.
This is used when the set continues without end in the positive direction on a number line. Example: [ a, ∞)
Negative Infinity - the -∞ symbol is used to denote negative infinity.
This is used when the set continues without end in the negative direction on a number line. Example: (-∞, a]