Set Theory Flashcards
What are the 3 types of statements?
Universal, Conditional and Existential
What is a universal statement?
A universal statement says that a certain property is true for all the elements in the set.
for each, for all
e.g., All positive numbers are greater than zero
What is a conditional statement?
A conditional statement says that if one thing is true, then some other thing also has to be true
If p then q
e.g., If 378 is divisible by 18, then 378 is divisible by 6
What is an existential statement?
An existential statement says that there is at least one thing for which a property is true, given that a property may or may not be true.
for some, there exists
e.g., There is a prime number that is even
Give an example of a universal conditional statement.
For all animals a, if a is a dog, then a is a mammal.
Coming from:
COND.: if an animal is a dog, then the animal is mammal
UNI.: all dogs are mammals
Give an example of an existential universal statement.
There is a positive integer that is less than or equal to every positive integer.
What are the following sets called: R Q Z N
R - set of all real numbers
Q - set of all rational numbers
Z - set of all integers
N - set of all natural numbers (positive)
Read this in words:
{x ∈ S | P(x)}
The set of all elements x in S such that P(x) is true.
If A and B are sets, then A is called a subset of B if and only if …
… every element of A is also and element of B.
If A and B are sets, then A is called a proper subset of B if and only if …
… every element of A is in B, but there is at least one element in B that is not in A.
What is a Cartesian Product?
Given the sets A and B, the Cartesian product of A and B, denoted A x B (read cross), is the set of all ordered pairs (a,b), where a is in A and b is in B.
What is an ordered pair?
Given 2 elements a and b, (a,b) denotes the order pair consisting of a and b together with the specification that a is the first element of the pair and b is the second element.
When are two sets equal to each other?
A = B if and only if every element of A is in B and every element of B is in A.
The order in which the values appear in the set is immaterial.
Define A U B (union)
The union of A and B is the set of all elements that are at least in one of A and B.
Define A ∩ B (intersection)
The intersection of A and B is the set of all elements that are common to both A and B.
Define the difference of B and A (also called relative complement) B - A.
The difference of B minus A is the set of all values that are in B but not in A.
Define the complement of A (denoted Aˆc)
The complement of A is the set of all elements in the universal set U, that are not in A.
What is an empty set? Ø
The only set with no elements. Also called the null set.
When are two sets disjoint?
Two sets are disjoint when they have no elements in common.
Which mean, A ∩ B = Ø
When is a finite or infinite collection of nonempty sets {A1, A2, A3, …} a partition of a set A?
A finite or infinite collection of nonempty sets {A1, A2, A3, …} is a partition of a set A if an only if:
- A1, A2, A3 … are mutually disjoint
- And the union of A1, A2, A3… equals A
What is a power set?
Given a set A, the power set of A is the set of all subsets of A.
Ø cannot be subset of a set. True or false?
False, Ø is a subset of a every set
What is the cardinality of a set?
The cardinality of a set is the number of (distinct) elements which it contains.
What cardinality does an infinite set have?
Infinite cardinality
What is the cardinality of Ø
0
