Set Theory

Question 1

What is a Set?

Answer 1

A set is a collection of distinguishable objects.

Question 2

How are the objects contained in a set called?

Answer 2

They are called members or elements

Question 3

How can you specify that an element is in a set?

Answer 3

x \in S

Question 4

How can you specify that an element is not in a set?

Answer 4

x \notin S

Question 5

Can the set contain an element more than one time?

Answer 5

No, it cannot

Question 6

How is called the variation of the set that can contain an object multiple times?

Answer 6

Multiset

Question 7

Are the elements (also called members) of a set ordered?

Answer 7

No, they are not

Question 8

When two sets are equal?

Answer 8

When they contains the same elements.

Question 9

What is an empty set?

Answer 9

A set that does not contain any element

Question 10

What is a singleton set?

Answer 10

A set containing exclusively one element

Question 11

What does it mean that the set A is a subset of B?

Answer 11

If all the elements of A are contained in a set B

Question 12

What is a proper subset?

Answer 12

If all the elements of A are contained in a set B, but the two sets A and B are not equal

Question 13

Is it true that any set is the subset of itself?

Answer 13

Yes

Question 14

How can we mathematically indicate that two sets are equal?

Answer 14

A = B \iff A \subseteq B \land B \subseteq A

Question 15

Is it true that for any set A, the empty set is included?

Answer 15

Yes, indeed: \forall A, \emptyset \subseteq A

Question 16

Can we define sets in terms of other sets?

Answer 16

Yes

Question 17

What are the set operations?

Answer 17

• Intersection
• Union
• Difference

Question 18

What is the intersection operation?

Answer 18

A \cap B = { x : x \in A \land x \in B }

Question 19

What is the union operation?

Answer 19

A \cup B = { x : x \in A \lor x \in B}

\lor indicates an inclusive or

Question 20

What is the difference operation?

Answer 20

A - B = { x : x \in A \land x \notin B }

Question 21

What is the intersection of the set A with the empty set?

Answer 21

The empty set

Question 22

What is the union of the set A with an empty set?

Answer 22

The set A

Question 23

How to name the laws describing?

• The intersection of the set A with the empty set
• The union of the set A with the empty set

Answer 23

Empty set laws

Question 24

What are the idempotency laws?

Answer 24

A \cup A = A
A \cap A = A

Question 25

Apply the commutative laws to the two given sets A and B

Answer 25

A \cup B = B \cup A A \cap B = B \cap A

Question 26

Apply the associative laws to the two given sets A and B

Answer 26

A \cap (B \cap C) = (A \cap B) \cap C A \cup (B \cup C) = (A \cup B) \cup C

Question 27

Apply the distributive laws to the two given sets A and B

Answer 27

A \cup (B \cap C) = (A \cup B) \cap (A \cup C) A \cap (B \cup C) = (A \cap B) \cup (A \cap C)

Question 28

Apply the absorption laws to the two given sets A and B

Answer 28

A \cap (A \cup B) = A A \cup (A \cap B) = A

Question 29

Apply the DeMorgan's Laws to the two given sets A and B

Answer 29

A - (B \cap C) = (A - B) \cup (A - C) A - (B \cup C) = (A - B) \cap (A - C)

Question 30

When two sets are disjoint?

Answer 30

When they do not have any elements in common It mathematically means: A \cap B = \emptyset

Question 31

How a partition of the set S is defined?

Answer 31

A collection C of nonempty sets is defined a partition if: - The sets are pairwise disjoint - The unions of all the sets of the collection C compose the set S

Question 32

How the cardinality (also called the size) of a set is defined?

Answer 32

The cardinality of a set is the number representing the number of elements (also called members) composing the set

Question 33

What are the laws to which the set operations obey?

Answer 33

- Empty Sets laws - Idempotency laws - Commutative laws - Associative laws - Distributive laws - Absorption laws - DeMorgan's laws

Question 34

How to define the complement of the set A

Answer 34

A^c = U - A = \{ x : x \in U \land x \notin A \}

Question 35

How do we define a set whose cardinality is a natural number?

Answer 35

Finite set

Question 36

How do we define a set whose cardinality is not a natural number?

Answer 36

Infinite

Question 37

What types of infinite sets do exist?

Answer 37

- Countable infinite - Uncountable infinite

Question 38

What does it mean for a set to be countable finite?

Answer 38

The set could be put in a one-to-one correspondence with the \mathbb{N} or \mathbb{Z}

Question 39

What does it mean for a set to be uncountable infinite?

Answer 39

The set could not be put in a one-to-one correspondence woth \mathbb{N} or \mathbb{Z}

Question 40

What is the relationship of cardinality with union of two sets?

Answer 40

|A \cup B| = |A| + |B| - |A \cap B|

Question 41

Is the triangle inequality applied to the cardinality of sets?

Answer 41

|A \cup B| \leq |A| + |B|

Question 42

How is called a set of n elements?

Answer 42

n-set

Question 43

How is called a set of 1 element?

Answer 43

Singleton

Question 44

How is called a subset of k elements of a generic set?

Answer 44

k-subset

Question 45

How do we call the set of all the subsets of a set S?

Answer 45

Power set

Question 46

How many elements there are in a Power Set of a set S containing n elements?

Answer 46

2^n or 2^{|S|}

Question 47

What will be the cardinality of a power set built over a set S having n elements?

Answer 47

2^n or 2^{|S|}

Question 48

How to the define the ordered pair (a, b) according to the set theory?

Answer 48

(a, b) = \{ a, \{ a, b \} \}

Question 49

How do we define the Cartesian product of two sets A and B?

Answer 49

The Cartesian Product of the two sets A and B is defined as the set of all the ordered pairs such that the first element of the pair is an element of A and the second element of the pair is an element of B A \times B = \{ (a,b) : a \in A \land b \in B \}

Question 50

What is the cardinality of the cartesian product of two sets?

Answer 50

|A \times B| = |A| \cdot |B|

Question 51

How the Cartesian Product of n sets is defined?

Answer 51

The Cartesian Product of n sets is the set of n-tuples A_1 \times A_2 \times \ldots \times A_n = \{ (a_1, a_2, \ldots, a_n) : \forall a_i \in A_i, i = 1, 2, \ldots, n \}

Question 52

What is the cardinality of the Cartesian Product of n sets?

Answer 52

|A_1 \times A_2 \times \ldots \times A_n| = |A_1| \cdot |A_2| \cdot \ldots \cdot |A_n|

Question 53

How is defined A^n?

Answer 53

A^n = A \times A \times \ldots \times A

Question 54

How to define the cardinality of A^n

Answer 54

|A^n| = |A|^n, if A is finite