CP 10 Set Cardinality

Question 1

what is the definition of cardinality

Answer 1

A and B are sets, if there exists a bijection from A to B then we say A and B have the same cardinality

Question 2

what the notation of cardinality

Answer 2

for every integer n≥1 let [n] denote the set {1,2,3,4…,n}

Question 3

TF [0]={}

Answer 3

T

Question 4

Let S= {choco chip, PB, oatmeal, shortbread}
consider f: S–>[4]
what is the cardinality of S

Answer 4

|S|=4

Question 5

how does the cardinality of a powerset work

Answer 5

if [3]=1,2,3 then
2^[3] = {{},{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

Question 6

what are binomial coefficents

Answer 6

written as (n/k) with no division sign, It represents the number of ways to choose k items from a pool of n items

Question 7

whats the formula for a binomial coefficent

Answer 7

n! / k!(n−k)!

ex) (5choose2)
5! / 2!(5-2)! = 10​

Question 8

TF n choose k = |[n]k|

Answer 8

T

Question 9

examples of binomial coefficents [5]2 and [5]3

note:(5choose2) = (5choose3)

Answer 9

[5]2 = (1,2) (2,3) (3,4) (4,5) (1,3) (2,4) (3,5) (1,4) (2,5) (1,5)

[5]3 = (1,2,3) (1,3,4) (1,4,5) (2,3,4) (2,4,5) (1,2,4) (1,3,5) etc

Question 10

Fact

Answer 10

Given n and k where n≥k≥1
(nchoosek) = (n-1choosek)+(n-1 choose k-1)

Question 11

TF (0 choose 0) = 1

Answer 11

T

Question 12

what is a combinatorial proof

Answer 12

we try and create a bijection

Question 13

what is the binomial theorem

Answer 13

let x be a variable n∈ℕ then,
(1+x)^n=n∑k=0 (nchoosek)

Question 14

what is an n factorial

Answer 14

n!
ex) 1! = 1
2! = 2x1! = 2
3! = 3x2! = 6
4! = 4x3! = 18

Question 15

TF 0! = 0

Answer 15

F, 0! = 1