Set Theory Flashcards
What is a set?
An unordered collection of objects of same type, called elements/members of set. 2 sets are equal if they contain exactly same elements, no matter the order. Size (cardinality) of set given by card (card ({2,3,5}) = 3. If set contains multiple copies of same element, only counted once. e.g. EvensToTen = {0,2,4,6,8} A set can be empty {}.
How do we show that something is/isn’t an element/member of a set?
p ∈ EvensToTen if P = 4
p ∉ EvensToTen if P = 3
What are 3 standard sets?
Z is set of all integers, N is set of all natural nums + R is set of all real nums.
What are 3 ways we can manipulate sets?
Intersection, union + difference.
What is the symbol for intersection + what does it do?
A∩B gives elements that are in both sets.
What is the symbol for union + what does it do?
A∪B gives elements that are in either set.
What is the symbol for set difference + what does it do?
A\B gives elements in A but not B.
How do we define a more complex set?
Start with big, easy to define set + throw away values we don’t want. Starting set called U (universal) + give conditions for values we want to keep.
Give the notation for defining more complex sets.
{x | x ∈ U . x condition }
e.g. x | x ∈ {1,…,10} . x mod 2 = 1} = {1,3,5,7,9}
What are operators for ‘and’ and ‘or’ in more complex sets?
and = ^ (like intersection) or = v (like union)
What is the value between { and | known as?
It is the typical member of the set and can be an expression rather than just a single variable. If every value from universal set is to be used, the condition can be omitted.
Can we use multiple variables in complex set definitions?
Yes. Just create an expression as normal and use a comma to separate them after the |
How do we create sets where elements are pairs of numbers?
like this: {[x,y]|x,y ∈ {0,…,10}.x+y=10}
How would we get a set of pairs and remove duplicates?
{[x,y]|x,y ∈ {0,…,10}.(x+y=10) ^ x<=y}
How would we find the number of pairs which result from a set comprehension?
card {[d,f]|d∈D,f∈F}
