1- Set Theory

Question 1

What is a set?
- Is the order relevant?
- The elements are specified between…
- ∈ represents…

What is a set builder notation?

Answer 1

Collection of distinct objects called elements
- the order is irrelevant
- between curly brackets
- an element belongs to a set

• notation that specify a criteria rather than listing all the elements of a set.

Question 2

N, Z, Q, R

Answer 2

N: set of all the natural numbers
Z: set of all the integers
Q: set of all the rational numbers
R: set of all real numbers

Question 3

A⊄B;
A⊆B;
A ⊂B;
A=B;

Answer 3

A∉B: at least one element of A does not belong to B;
A⊆B A subset of B: all the elements of A belong to B;
A ⊂B A proper subset of B: if A⊆B and there is at least 1 element of B that does not belong to A;
A=B A equal to B: if they contain the same elements.

Question 4

• 1 ∈ {1,2,3}?
• 1 ⊆ {1,2,3}?
• {1} ∈ {1,2,3}?
• {1} ⊆ {1,2,3}?

Answer 4

• yes
• no
• no
• yes

Question 5

Ordered Pair:
- what is it?
- how is it indicated?
- what is a n-tuple?

Answer 5

• ordered pair of values where the order matters;
• (a,b);
• ordered pair with n elements.

Question 6

Cartesian Product of set A and B:
- how is it indicated? How do you say ‘x’?
- what is it?
- notation

Answer 6

• AxB: ‘A cross B’;
• the set of all ordered pairs (a,b) where a ∈ A and b ∈ B;
• AxB = {(a,b) | a ∈ A and b ∈ B}

Question 7

Venn diagram:
- what does it show?

Answer 7

• it shows the logical relation between sets.

Question 8

If A and B are subsets of a universal set U:
- AuB
- AnB
- B\A
- A^c

Answer 8

• The union of A and B is the set of elements from both sets {x∈U | x∈A or x∈B};
• The intersection of A and B is the set of common elements from both sets;
• the difference of B minus A is the set of elements in B that are not in A.
• the complement of A is the set of all the elements except elements in A.

Question 9

• What is an empty set? How is it represented?
• When are 2 sets disjoint?
• When are n sets mutually disjoint?

Answer 9

• A set with no elements, o/

-Given 2 sets, they are disjoint if they have no elements in common (their intersection is the empty set).

• Given n sets, they are mutually disjoint if each pair of sets is disjoint (Ai n Aj = o/ for each i != j).

Question 10

What is a partition of a set? (2 requirements)
- make an example

Answer 10

X is a partition of A if X is a collection of non empty sets {A1, A2..} such that:
- the union of all the sets of the collection is equal to A;
- the sets in the collection are mutually disjoint.

Question 11

Power Set of A:
- how is it indicated?
- what is it?
- what is its cardinality and why?
- make example, is o/ included? is A included?

Answer 11

• P(A)
• the set of all subsets of A;
• 2^n: each element of A can be either included or not in the subsets.
• A= {1,2} then P(A) = {o/, {1},{2},{1,2}}

Question 12

What is the cardinality of a set? How is it indicated?

Answer 12

The cardinality of a set A, |A|, is the number of elements that set A contains.

Question 13

3 types of statements:
- list them and state the symbols
- make examples

• mix them

Answer 13

• universal: a property true for all elements in a set (for all, for each, V).
  ‘all positive numbers are greater than zero’.
• conditional: if one thing is true then some other thing has to be true (if p then q).
  ‘if month is February then days less than 30’.

-existential: given a property, there is at least one thing for which the property is true (there exists, there is one, for some, E).
‘there is a prime number that is even’

• universal conditional: ‘for every year, if month = February then days <30’
• universal existential: ‘for each real number x, there exists a real number y such that x+y = 0.