Discrete Structures MidTerm1

What are the Natural Numbers

{1,2,3,4....}

Suppose A1 = {a,b,d, e, g, f } , A2 = { a,b, c,d } , A3 = {b,d,a } and A4 = { a,b,h } . Find: (top 4) ∩ (bottom i=1)

{a,b}

Suppose A1 = {a,b,d, e, g, f } , A2 = { a,b, c,d } , A3 = {b,d,a } and A4 = { a,b,h } . Find: (top 4) ∪ (bottom i=1)

{abcdefgh}

What is the (not)Subset Symbol

⊄ & ⊆

Let A = {a,b,c,d,e} b={d,e,f} and C = {1,2,3} Find: (A∪B) (A-B)∪(B-A) (A∩B)∪(A-C)

(A∪B)= {a,b,c,d,e,f} (A-B)∪(B-A)= {a,b,c}∪{f}= {a,b,c,f} (A∩B)∪(A-C)= ∅∪{a,b,c,d,e}={a,b,c,d,e}

For each n ∈ N, let An = {−2n,0,2n} find: (a) (bottom i∈N) ∪ Ai = (b) (bottom i∈N) ∩ Ai =

(a){-2,0,2},{-4,0,4},... (b){0}

Solve(2.8) (a) (bottom X∈P(N)) ∪ X = (b) (bottom X∈P(N)) ∩ X =

(a){N} (b){∅}

Are these subsets and elements: 1 ⊆ {1,{1}} 1 ∈ {1,{1}} N ∈ N N ⊆ N {{1}} ∈ {1,{1}} {{1}} ⊆ {1,{1}}

1 ⊄ {1,{1}} 1 ∈ {1,{1}} N ∉ N N ⊆ N {{1}} ∉ {1,{1}} {{1}} ⊆ {1,{1}}

How many lists of length 3 can we make with the 1st entry being (a,b,c) the 2nd entry is {5,7} and the 3rd is {a,x}

12 (a,5,a)(a,5,x)(a,7,a)(a,7,x)(b,5,a)(b,5,x)(b,7,a)(b,7,x)(c,5,a)(c,5,x)(c,7,a)(c,7,x)

What are the subsets of {1,2,3}

8 subsets: {}, {3}, {2}, {1}, {2,3}, {1,3}, {1,2}, {1,2,3}

What is A with a bar over it

A compliment

Let: A={0,2,4} B={1,3,5} U={1,2,3,4,.....7} What are A and B compliment

A compliment= {1,3,5,6,7} B compliment= {0,2,4,6,7}

What are the Real Numbers

Any number(e.g. 2.5,6.27, 0, 10)

When can you use the addition principle

If there is no overlap

Venn diagram for (A∪B)∩C

Middle part of all three

(power set) P(P({1}))

P(P({1})) = p({∅,{1}}) = {∅ {∅,{{1}} {∅,{1}}

What is the notation for permiatations

P(n,k)

(power set) P({a,1,5})

P({a,1,5}) ={∅, {a}{1}{5}, {a,1}{a,5}{1,5}, {a,1,5}

What does this sign mean and when is is false: ⊕

Px or Q false when both a true and both are false(reverse of ⇔) PQ P ⊕ Q TT F TF T FT T FF F

What does the notation P(n,k) stand for

The number of k-permutations of an n-element eg. P(4,2) = 12 (4!/4!-2! = 4*3)

What does the subtraction principal say

U(universe)-X = ANSWER

Union of a and b(A∪B)

all of a and b

How to solve nested summation and product notations

do the inside one first them put that result in the outer one eg: top(4) bottom(i=2) Σ i^2 top(3)bottom(j=1) ∏ (i+j) = (i+1)(i+2)(i+3) = top(4) bottom(i=2) Σ i^2= (i+1)(i+2)(i+3) =(i^3+6i^2+11i+6)=(2^3+6(2)^2+11(2)+6)+(3^3+6(3)^2+11(3)+6)+(4^3+6(4)^2+11(4)+6) =60+120+210=390

What is the difference between a Element and a Subset

element is a part of a set x∈A, while a subset is a set containing elements in the set

What does "∈" mean

element of

What does this sign mean and when is is false: ~

it is not true that P false when p is true

When do you do p(n,k)

most likely it will say LINEUPS

What does (n k) equal

n!/k!(n-k)!

What does this sign mean and when is is false: ∨

or(reverse of and) false when both are false

Venn diagram for (A∩B)∩C

tiny center part of all three(where they all overlap)

Intersection of a and b(A∩B)

what a and b share

How do you know if a question is asking about stars and bars

when it asks for a MULTISET or is is like z+y+x+w = #

How do you find AxB or the Cartesian Product

The sqaure thing, eg.: A={k,l,m} B={2,10} AxB= {(k,10)(k,2)(l,10)(1,2)(m,10)(m,2)} ***A is always on the bottom and 1st

What does the addition principal say

a bunch of tiny sets can equal to one big set, so X1+X2+X3....Xn = ANSWER

what is a permutation

a nonrepetitive list

What do this sign mean and when is is false: ^

and(reverse of or) false only when both are not true So only true when both are true

Difference of a and b(A-B)

everything in a that is NOT in b

What do this sign mean and when is is false: ⇔

if and only if False when both are not the same(reverse of ⊕) PQ P ⇔ Q TT T TF F FT F FF T

What does this sign mean and when is is false: ⇒

if p then q False only when P is true and Q is false

How many sets are in a power set

if the set has 2 values then there are 0,1,2 sets, if 3 then 0,1,2,3, and so on of those sizes, 2^|A| if finite, eg: P({b,c,8}) ={∅, {b}{c}{8}, {b,c}{b,8}{c,8}, {b,c,8} {set of 0, set on 1's, set of 2's, set of 3}

How to express the Product of a set(eg 2k+1 starting from 3 ending at 5)

top(5)bottom(k=3) ∏ (2k+1) =(2*3+1)(2*4+1)(2*5+1)=7*9*11=693

How to express the Sum of a set(eg i^2 starting from 1 ending at 6)

top(6) bottom(1) Σ i^2= 1^2+2^2+3^2+4^2+5^2+6^2=91

How does stars and bars work

you count the stars which is the amount in the total set, then count the bars which is the space between each number in the set, then add the total ex. 5 stars, 3 bars, 8 total, so (8 5) or (8 3)

What are Integers

{...-3,-2,-1,0,1,2,3...}

Are these or are these not subsets {1,3} of {0,1,2,3} {1,3} of {0,1,2} {1,3} of {3,1} {1,3} of {Ø} {Ø} of {1,3}

{1,3} ⊆ {0,1,2,3} {1,3} ⊄ {0,1,2} {1,3} ⊆ {3,1} {1,3} ⊄ {Ø} {Ø} ⊆ {1,3}

T={a,{b}} {b} ⊆ T {b} ∈ T {a} ⊆ T {{b}} ⊆ T

{b} ⊄ T {b} ∈ T {a} ⊆ T {{b}} ⊆ T

Notation for a set

{expression:rule} eg. {n: n is an even integer} {n: n=2k where k=2}

What does "or" indicate you using/doing

|A|+|B|-|A ∩ B|

Inclusion-Exclusion Formula

|A|+|B|-|A∩B|

What is the set: {2x: x ∈ Z and |x|<4}

|x|<4 gives you -3 through 3, so put these answers in 2x to get {-6,-4,-2,0,2,4,6}