Precalculus (6)

subsets

If every element in A is also in B, then A is a subset of B

A ⊆ B

Disjoint/mutually exclusive sets

No intersection between a set A and a set B; nothing in common

Intersection

Set that includes members of both A and B

A ∩ B

Union

Set that includes members of A, B, or both

A U B

The Complement

set of elements not in A

Ā or A^c

What is meant by:

in A or B

A and B

At most one?

• Anything in A, B, AB, and AC because the problem doesn’t specifically rule it out

• Only elements in A ∩ B

• Think logically here, could be in 1 or none

when writing answers…

write the factorial as work, and then have the integer value as the circled answer!

when are there multiple steps and how do you go about solving them…

(confirm once you prove this)

If they say “at most’ or any other phrase that indicates you have to first eliminate options and choose amongst the remaining

When dealing with sets…

Don’t forget those in neither of the sets and the fact that if there’s no restriction, you must think of every possible element that can fit the criteria!

“Every element not in A or C” —> Can be in just B, or none

When approaching arrangement or selection problems…

read the whole question and underline important phrases/adjectives that indicate either permutation or combination

“elect 5 members to a council” - nCr

“elect 5 members to a different responsibility..” - nPr

explain nPr

• Permutations (arrangements); n!/(n-r)!

• n = total elements

• r = “slots” to distribute to (if any)

• n-r = the excess factorial

explain nCr

• Combinations (selections); n!/(n-r)!r!

• r! = since ABC and CBA are the same, must factor out duplicate selections

explain the result of combining permutations and combinations (formula)

• Distinguishable permutations

• n!/(n-r)!x!y!z!

• (n-r)! = excess factorial (if any)

• x,y,x,etc. = repeats of a certain element, factors out repeated arrangements

  • bomb = 4!/2! ←2 b’s

  • bomb in 2 slots = 4!/2!2! ←the 2 slots + the 2 b’s

• Recognize: aabbbccc

explain (n-1)!

• Circular permutations

• in order to determine the amount of ways to rearrange elements in a circle, you must do it in respect to one of the elements, and then rearranging the remaining

explain (n-1)!/2

• Reflexive circular permutations

• When dealing with objects that are arranged in a medium that can be flipped, therefore making half of the possible arrangements the same.

If out of intersecting sets A, B, and C you are asked for items in A and B, what does that include?

• Items in the union of A and B, and in the union of A, B, and C as since there was no restriction on what couldn’t be included, assume that the questions asks for everything that fits the criteria

  • This also fits for situations were you are asked at most # categories, which also includes those who chose none!

• Just remember to circle “and” and “or” on the test as well as “at most/least”

When taking the test…

Don’t overcomplicate anything!

Write what you know, move in and out of problems if stuck

mistake with 2 part problems

make sure you’re only apply the restrictions to the part their specifically assigned to!

If there are part-specific restrictions, solve for each side and combine at the end

mistake with combinations

don’t forget them! Constantly question whether, in a part of a problem, if you are dealing with selections or arrangements!

what do “and” and “or” signify in permutations/combinations?

or = tends to be adding two possibilities (for example, in the ‘equation’ problem, each qu arrangement was “complete” and therefore will be added to each other and not *)

and = tends to be multiplying two parts of a possibility

circle them!