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common. The lowest of these is known as the LCM, or lowest common
multiple. To find the LCM more efficiently, each number can be again broken
down into the product of its prime factors. Then, the prime factors of one of the
numbers that are NOT prime factors of the other numbers should be taken
out. These should be multiplied by the originally chosen number. This yields
the same value regardless of which number is originally chosen, and this is
the LCM.
Sets
Sets are collections of elements
Elements in a set can be any individual piece of information, numbers, letters,
coordinates, etc.
A set can be defined by writing its elements inside of ‘{}’ brackets. E.g. {1, 2, 4,
8} is the set of the factors of 8
If the elements in a set follow a rule, a colon can be used to separate the
general type of element from the rule. E.g. in the set {x: x2 < 50}, the elements
are the numbers that are less than fifty when squared. Coordinates and their
corresponding equations can also be represented this way, e.g. {(x, y) : y = 3x
- 2} defines the points that lie on the line y = 3x - 2
Set Representation
Uppercase letters (e.g. A, B, C) usually represent sets, whereas lowercase
letters (e.g. a, b, c) represent general elements
The symbol ε or U denotes the universal set (set of everything). E.g. if the
subject is the factors of 12, ε = {1, 2, 3, 4, 6, 12}
The symbol ∅ denotes the empty set (with no elements). E.g. {x : x is an even
prime bigger than 2} = ∅ as there are no even primes bigger than 2
Algebraic Operations of Sets
n(A) represents the number of elements in the set A. E.g. n{-1, 0, 1} = 3
a ∈ A means a is an element of A. E.g. if E is the set of even numbers, 24 ∈ E
A ⊂ B means set A is a proper subset of a larger set B. E.g. if X = {1, 2, 3} and
Y = {0, 1, 2, 3, …}, then X ⊂ Y
A ⊆ B means that set A is a subset of set B, which has the same elements,
but does not necessarily have to be larger. E.g. if N = {2, 4, 6}, then {2, 4, 6} ⊆
N
Putting a cross through the ∈, ⊂, or ⊆ signs renders them untrue, similar to
the = sign. So a ∉ A, for example, means that a is not an element of A