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Discrete Mathematics > Set Theory > Flashcards

Set Theory Flashcards

1
Q

What is a Set?

A

A set is a collection of distinguishable objects.

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2
Q

How are the objects contained in a set called?

A

They are called members or elements

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3
Q

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

A

x \in S

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4
Q

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

A

x \notin S

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5
Q

Can the set contain an element more than one time?

A

No, it cannot

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6
Q

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

A

Multiset

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7
Q

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

A

No, they are not

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8
Q

When two sets are equal?

A

When they contains the same elements.

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9
Q

What is an empty set?

A

A set that does not contain any element

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10
Q

What is a singleton set?

A

A set containing exclusively one element

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11
Q

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

A

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

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12
Q

What is a proper subset?

A

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

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13
Q

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

A

Yes

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14
Q

How can we mathematically indicate that two sets are equal?

A

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

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15
Q

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

A

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

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16
Q

Can we define sets in terms of other sets?

A

Yes

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17
Q

What are the set operations?

A
  • Intersection
  • Union
  • Difference
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18
Q

What is the intersection operation?

A

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

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19
Q

What is the union operation?

A

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

\lor indicates an inclusive or

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20
Q

What is the difference operation?

A

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

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21
Q

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

A

The empty set

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22
Q

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

A

The set A

23
Q

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
A

Empty set laws

24
Q

What are the idempotency laws?

A

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

25
Q

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

A

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

26
Q

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

A

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

27
Q

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

A

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

28
Q

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

A

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

29
Q

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

A

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

30
Q

When two sets are disjoint?

A

When they do not have any elements in common

It mathematically means: A \cap B = \emptyset

31
Q

How a partition of the set S is defined?

A

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
32
Q

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

A

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

33
Q

What are the laws to which the set operations obey?

A
  • Empty Sets laws
  • Idempotency laws
  • Commutative laws
  • Associative laws
  • Distributive laws
  • Absorption laws
  • DeMorgan’s laws
34
Q

How to define the complement of the set A

A

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

35
Q

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

A

Finite set

36
Q

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

A

Infinite

37
Q

What types of infinite sets do exist?

A
  • Countable infinite
  • Uncountable infinite
38
Q

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

A

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

39
Q

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

A

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

40
Q

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

A

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

41
Q

Is the triangle inequality applied to the cardinality of sets?

A

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

42
Q

How is called a set of n elements?

A

n-set

43
Q

How is called a set of 1 element?

A

Singleton

44
Q

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

A

k-subset

45
Q

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

A

Power set

46
Q

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

A

2^n or 2^{|S|}

47
Q

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

A

2^n or 2^{|S|}

48
Q

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

A

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

49
Q

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

A

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 }

50
Q

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

A

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

51
Q

How the Cartesian Product of n sets is defined?

A

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 }

52
Q

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

A

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

53
Q

How is defined A^n?

A

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

54
Q

How to define the cardinality of A^n

A

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

Discrete Mathematics (1 decks)

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