Sets and Subsets
Example with Power Sets (set of all subsets of A)
A = { 2, 3, 5, 7, 14 } > is 2 ∈ P(A)? no, because it should be {2}, 2 ∈ A instead > { 2, 3 } ∈ P(A).? yes, because that would be of size 2 contained in the power set REMEMBER: P(A) = {{2}, {3}, {4}, {5}, {6}, {2,3}, {2,4}, {2,5}, {3,4}, {3,5}, {4,5} ...}
Unions and Intersections of Sequences of Sets
4.3.7 Image... > Basically if we're dealing with the oxford dictionary, we can use unions and interactions to make statements about similarities/differences in the words in the dictionary > 45 ∪ j=1 Aj would represent all words in the OED because we would be summing all words 1 through 45
Symbols to Sets
∈ = element is in a set ∉ = element not in a set ∅ = empty set |A| = cardinality (# of elements) of a set A if and only if it is finite (terminates at a certain value) ∅ = { } is an empty set - keep in mind that 0 is an integer N = natural #s (0 and greater) Z = all integers Q = rational #s (real numbers that can be expressed as a / b) R = real numbers (every number including negative, positive, 0, square roots, fractions)
Cartesian Products of Many Sets
- EX: A = { 1, 2, 3 } B = {x, y} C = {u, v, w} D = { +, *} 1) (w, y, 2) ∈ C x B x A > we can determine if this is true or false by looking at the corresponding values. In C, there is a w in the tuple. In B, there is a y in the tuple. In A, there is a 2 in the tuple, therefore this Cartesian product is TRUE. 2) (1, x, u, +) ∈ B x A x C x D > we can determine if this is true or false by looking at the corresponding values. B does not have 1 contained in the tuple, instead it has {x,y}, therefore we can conclude this is FALSE
Difference of Sets (-)
- EX: A - B > set of elements that are in A but not in B (elements that are in both A and B are also excluded)
Partition
- a small fraction of the total amount of elements in the set EX: A = {1,2,3,4,5} Partition of A: A1 = {1,2} A2 = {3} A3 = {4,5}
Union (∪)
- all elements in both sets
Proper Subset (⊂)
- an element of B that is not an element of A denotes A as a proper subset of B > A ⊂ B > This means that every element in A is in B but every element in B is not in A
Lambda (λ) when dealing with strings
- an empty string that does not affect string size (blank) EX: xλ = x
Curly Braces vs Parentheses
- curly braces when dealing with elements in a set, order does not matter - when dealing with ordered pairs (x,y) order matters, so (x,y) is not equivalent to (y,x)
Disjoint and Pairwise Disjoint
- disjoint : A ∩ B = ∅, meaning no similarities between the two sets - pairwise disjoint/mutually disjoint : Ai ∩ Aj = ∅ for any i and j in the range from 1 through n where i ≠ j
Sets
- does not have to be a certain order - repeated elements does not affect set
Complement (A with line on top)
- elements in the set "U" that are not elements of A - means "not", so the opposite - basically, if we're visualizing this as a venn diagram, it's the opposite of what is shaded > EX: U = {1,2,3,4,5} > A = {1,2,3} > A w/lineTop = {4,5} > EX: if we have just A and B both with lines on top, this would mean all elements that aren't included in A and B
Intersection (∩)
- elements similar in both sets
Symmetric Difference (A ⊕ B)
- elements that are contained in A and B but not in both sets
Cartesian Product of a similar set
- if A x A, we can say A^2
Subset (⊆)
- if A ⊆ B, this means that A is a subset of B (every element in A is also an element of B - if A ⊈ B, this means that there is an element of A that is not an element of B
Positive vs Negative vs Non-negative
- positive means that x > 0 - negative means that x < 0 - non-negative means that x >= 0
|A x B| = 5
- remember | | means the total amount of elements
Universal Set (U)
- set that contains all elements mentioned in a particular context EX: discussion about academic standing of certain students... the universal set would be the set of all students enrolled in the school
Equal Sets
- two sets are equal if and only if each set is a subset of the other > A = B if and only if A ⊆ B and B ⊆ A
Theorem: Cardinality of a Power Set
- |P(A)| = 2^n
Sets of Sets (section 4.2)
EX: A = { 3, 4, { 3, 4 }, { 1, 2, 3 }, 5 } > is { 3 } ⊆ A? > yes, because 3 is contained in the set A and also contained in the subsets {3,4} and {1,2,3} > is { 1, 2, 3 } ⊆ A? > false, because {1,2,3} is contained in one subset but not any other subset or set
A^n definition
EX: if we have A = {x,y} and we are asked to find A^2... it would be {xx, xy, yx, yy} EX: if we have A = {x,y,z} and we are asked to find A^3... it would be {xxx,xxy,xxz,xyx...} IT HAS TO HAVE 3 STRINGS!!!!