© Jill Zarestky
Math 141 Week in Review
Week in Review 5 Key Topics
6.1 Sets and Set Notation
A set is a well-defined collection of objects.
• The objects in a set are the elements or members of the set.
• Always enclose the elements of a set in curly brackets.
• To say that a is an element of the set S, write a ∈ S.
Roster notation: List the elements of the set.
Set-builder notation: Describe the set in terms of its properties.
Equality
• Two sets are equal (=) if they contain exactly the same elements.
• Order doesn't matter.
Subset: Set B is a subset of set A (written B ⊆ A) if every element in B is in A.
Proper Subset: Set B is a proper subset of set A (written B ⊂ A) if B ⊆ A and A ≠ B.
The empty set, ∅, is the set with no elements. ∅ is a subset of every set.
Universal set: The set from which all the members of other sets will be drawn. This will vary
based on the given situation.
A Venn Diagram is a visual representations of sets.
• A rectangle represents the universal set
• Circles are subsets of the universal set.
Given a set A and a universal set U, the elements that are in U and are NOT in A are called the
complement of A, notated AC.
Those elements that belong to both A and B are in the intersection of A and B, A ∩ B.
If two sets have no elements in common, that is A ∩ B = ! , then the sets are said to be disjoint.
Those elements that belong to A or B are in the union, A ∪ B.
DeMorgan’s Laws
• (A ∪ B)C = AC ∩ BC
• (A ∩ B)C = AC ∪ BC
© Jill Zarestky
Math 141 Week in Review
6.2 The Number of Elements in a Finite Set
The number of elements in set A is n(A).
Union Rule: n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
6.3 The Multiplication Principle
Multiplication Principle: The total number of ways to perform a series of tasks is the
product of the number of ways to perform each task.
6.4 Permutations and Combinations
Factorials:
• n! = n(n – 1)(n – 2) . . . (3)(2)(1)
• 0! = 1
Permutations: ORDER MATTERS!
• The number of permutations of n distinct objects taken r at a time is
n!
P ( n, r ) = n Pr =
( n ! r )!
•
Use simply n! if you are arranging all of the objects.
What if the n objects contain some that are identical?
• Count only the distinguishable permutations, i.e. the ones that look different.
• If there are n1 items of type 1 and n2 items of type 2 and ... nr items of type r, then the
n!
number of distinguishable permutations of the n = n1 + n2 +...+ n r items is:
n1 !n2 ! ... nr !
Combinations: ORDER DOES NOT MATTER!
• The number of combinations of n distinct objects taken r at a time is
n!
C ( n, r ) = n Cr =
r!( n ! r )!