ICM Unit 1 Review

*Vocabulary is cumulative. Look back to other Unit 1 subtopics if you don’t remember a concept!*

Sets and Set Operations

Vocabulary to Remember:

●set: a well-define collection of objects

●roster notation: ex. S = {1, 2, 3, …}

●set builder notation: ex. { x l x > 0 }

○l stands for “such that”

●element: an object in a set

○ex. January is an element of Set A → January A

●subset: every element in set A is also in set B

○if A is smaller than B, it is a proper subset

●union: the set of all elements in A or B or both → A B

●intersection: the set of what sets A and B have in common → A B

●empty set: contains no elements → { } or

○is a subset of every set

●universal set: the set of all elements → u

●complement: the set of all elements in the universal set that aren’t in Set A → AC

Examples:

  1. Let U = set of all senators in Congress, D = { x U l x is a Democrat}, R = { x U l x is a Republican}, and F = { x U l x is a female}. Write the following in words (venn diagrams may help).
  2. F R
  3. FC D
  1. Write the roster notation of whole numbers between 10 and 15.

The Number of Elements in a Finite Set

Concepts to Remember:

●notation: n(A) = the number of elements in Set A

●n(A B) = n(A) + n(B) - n(A B)

○note: if n(A B) = 0, then A and B are disjoint, meaning n(A B) = n(A) + n(B)

●n(A B C) = n(A) + n(b) - n(A B) - n(B C) - n(A C) + n(A B C)

Examples:

  1. Write the notation of elements in the finite Set A if A = {2,4,5,7}.
  2. Use the venn diagram to the right to find the following.
  3. n(A B)
  4. n(A B)
  5. In a survey of 85 sports enthusiasts:

●8 like football

●24 like golf

●14 like hockey

●10 like golf and football

●7 like football and hockey

●15 like golf and hockey

●2 like all three

  1. Draw a venn diagram.
  2. How many sports enthusiasts don’t like football, golf, or hockey?

The Multiplication Principle

Vocabulary to Remember:

●Multiplication Principle: there are A ways of doing something and B ways of doing another thing; there are A B ways of performing both actions

●tree diagram: a diagram that shows all the possible outcomes of an event

Examples:

  1. An ice cream shoppe offers three flavors of ice cream: vanilla, strawberry, and chocolate. The shoppe also offers three toppings: chocolate sprinkles, rainbow sprinkles, and gummy bears. Assuming a customer can only have one topping per ice cream flavor, and one ice cream flavor per order:
  2. Draw a tree diagram to display all the possible outcomes of a customer's order.
  3. Use the multiplication principle (and your venn diagram as a guide) to find how many combinations of ice cream and toppings are possible.

Permutations and Combinations

Vocabulary to Remember:

●permutation: an arrangement of a set of objects in a definite order

●n-factorial: n!

●P(n,r) = → the number of permutations of n different objects taken r at a time

●combination: an arrangement of a set of objects without regard to order

●C(n,r) = → the number of combinations of n different objects taken r at a time

Examples:

  1. A boy has 4 beads that are different colors: red, blue, white, and yellow. How many different ways can the beads be arranged in a row? Is this a permutation or a combination?
  2. In how many ways can an investor select 10 mutual funds for his investment portfolio from a recommended list of 30 mutual funds? Is this a permutation or a combination?
  3. In how many ways can a subcommittee of 15 be chosen from a Senate committee of 10 Democrats and 27 Republicans if all senators are eligible? Is this a permutation or a combination?