DTG-1

Question 1

What does Ø (the empty set) represent in discrete mathematics?

Answer 1

It represents a set with no elements, used when no values satisfy the given condition.

Question 2

What is the union of two sets A and B in discrete mathematics?

Answer 2

The union of A and B, written as
A∪B
A∪B, is a set containing all elements that are in A, in B, or in both.

Question 3

What is the intersection of two sets
A and B in discrete mathematics?

Answer 3

The intersection of A and B, written as
A∩B
A∩B, is the set of all elements that are in both A and B.

Question 4

What is the difference between two sets A and B in discrete mathematics?

Answer 4

The difference of A and B, written as A \ B or A - B, is the set of all elements that are in A but not in B.

Question 5

What is the negation of a statement in discrete mathematics, and how is it represented?

Answer 5

The negation of a statement reverses its truth value. If the statement is true, its negation is false, and if the statement is false, its negation is true. It is represented as ¬P or ∼P.

Question 6

What does De Morgan’s law say about the negation of a conjunction (P AND Q)?

Answer 6

The negation of a conjunction (P AND Q) is the same as the disjunction of the negations of P and Q. In other words,
¬(P∧Q)≡(¬P)∨(¬Q).

Question 7

What does De Morgan’s law say about the negation of a disjunction (P OR Q)?

Answer 7

The negation of a disjunction (P OR Q) is the same as the conjunction of the negations of P and Q. In other words,
¬(P∨Q)≡(¬P)∧(¬Q).

Question 8

What is the result of a conjunction (P AND Q) when one of the parts is false?

Answer 8

The conjunction P and Q is false if either P or Q is false, since both must be true for the conjunction to be true.

Question 9

What is the result of a disjunction (P OR Q) when both parts are false?

Answer 9

The disjunction P or Q is false only when both P and Q are false.

Question 10

When is a conjunction (P AND Q) true?

Answer 10

A conjunction P and Q is true only when both P and Q are true.

Question 11

When is a disjunction (P OR Q) true?

Answer 11

A disjunction P or Q is true if at least one of P or Q is true.

Question 12

What does the cardinality of a set represent?

Answer 12

The cardinality of a set represents the number of elements in the set, indicating its “size.”

Question 13

What is the Cartesian product of two sets?

Answer 13

The Cartesian product of two sets, A and B, is the set of all ordered pairs (a, b), where a is an element of A and b is an element of B, written as A × B.

Question 14

What is an n-tuple in the Cartesian product of sets?

Answer 14

An n-tuple in the Cartesian product is an ordered collection of n elements, where each element comes from a corresponding set in the product. For example, in A₁ × A₂ × … × Aₙ, an n-tuple is (a₁, a₂, …, aₙ) with aᵢ ∈ Aᵢ.

Question 15

what is a subset?

Answer 15

A subset is a set where every element is also contained in another set.

Question 16

Is the empty set a subset of every set? Why?

Answer 16

Yes, the empty set is a subset of every set because it has no elements, so it automatically satisfies the subset condition.

Question 17

How many subsets does a set with 4 elements have?

Answer 17

A set with 4 elements has
2^4=16 subsets.

Question 18

What is a powerset?

Answer 18

A powerset is the set of all subsets of a given set, including the empty set and the set itself.

Question 19

If a set A has 3 elements, how many elements does its powerset have?

Answer 19

The powerset of a set with 3 elements has
2^3=8 subsets.

Question 20

What is the relationship between a set and its powerset in terms of cardinality?

Answer 20

If a set has n elements, the cardinality of its powerset is
2^n, because each element of the set can either be included in or excluded from a subset.

Question 21

What is a relation?

Answer 21

A relation is a set of ordered pairs, where each element from one set is associated with an element from another set (or the same set).

Question 22

What is a reflexive relation?

Answer 22

A relation is reflexive if every element in the set is related to itself, meaning for all a in the set, the pair (a, a) is in the relation.

Question 23

What is a transitive relation?

Answer 23

A relation is transitive if whenever (a, b) and (b, c) are in the relation, the pair (a, c) must also be in the relation.

Question 24

What is a symmetric relation?

Answer 24

A symmetric relation is a relation where if one element is related to another, the second element is also related to the first. For example, if a is related to b, then b is related to a.

Question 25

What is an antisymmetric relation?

Answer 25

An antisymmetric relation is a relation where if one element is related to another and the second element is also related to the first, then the two elements must be the same. For example, if a is related to b and b is related to a, then a must equal b.

Question 26

What does a relation represented as a graph consist of, and how can it be used to determine properties such as reflexivity, symmetry, and transitivity?

Answer 26

A relation represented as a graph consists of a set of vertices (elements of the set) and directed edges (ordered pairs in the relation). Reflexivity is checked by verifying that every vertex has a self-loop. Symmetry is determined by checking if every directed edge (x, y) is paired with its reverse (y, x). Transitivity is verified if, whenever there are edges (x, y) and (y, z), there is also an edge (x, z).

Question 27

What are the three properties that define a partial order on a set?

Answer 27

A partial order on a set is a relation that is: Reflexive: Every element is related to itself (a ≤ a). Antisymmetric: If a ≤ b and b ≤ a, then a = b. Transitive: If a ≤ b and b ≤ c, then a ≤ c.

Question 28

What is the difference between a partial order and a total (or linear) order?

Answer 28

A partial order allows some pairs of elements to be incomparable (neither a ≤ b nor b ≤ a). A total order requires that every pair of elements is comparable (for all a, b in the set, either a ≤ b or b ≤ a).

Question 29

What is an example of a partial order in a real-world scenario?

Answer 29

An example of a partial order is the subset relation (⊆) on the power set of a set. For any subsets A, B, and C: Reflexive: A ⊆ A. Antisymmetric: If A ⊆ B and B ⊆ A, then A = B. Transitive: If A ⊆ B and B ⊆ C, then A ⊆ C.

Question 30

What is lexicographic order?

Answer 30

Lexicographic order is like dictionary order: compare the first elements of sequences, then the next, until a difference is found or the sequences end.

Question 31

What is an equivalence relation?

Answer 31

An equivalence relation is a relation that is reflexive, symmetric, and transitive.

Question 32

What does reflexive mean in an equivalence relation?

Answer 32

Reflexive means every element is related to itself: a∼a.

Question 33

What is a partition of a set, and how does it divide a set into disjoint subsets?

Answer 33

A partition of a set is a way of dividing the set into non-empty, disjoint subsets such that every element of the original set is included in exactly one subset. The subsets must be mutually exclusive and cover the entire set.

Question 34

What is an equivalence class, and how is it related to an equivalence relation?

Answer 34

An equivalence class is a subset of a set, where all elements in the class are considered equivalent under a given equivalence relation. If a relation is an equivalence relation, it satisfies three properties: reflexivity, symmetry, and transitivity. The equivalence class of an element includes all elements that are related to it by the equivalence relation.

Question 35

What does it mean for two integers to be congruent modulo n, and how is it related to their remainders when divided by n?

Answer 35

Two integers a and b are congruent modulo n (written as a ≡ b (mod n)) if their difference a - b is divisible by n. This means a and b leave the same remainder when divided by n.

Question 36

Can you give an example of equivalence classes from something like "people born in the same year"?

Answer 36

This could be people born in q1, q2, q3 and q4: reflexivity: A person is always born in the same quarter as themselves.Symmetry: If person a is born in the same quarter as person b, then b is also born in the same quarter as a.Transitivity: If a is born in the same quarter as b, and b is born in the same quarter as c, thenaa is also born in the same quarter as c.

Question 37

What are the congruence classes modulo 6, and what integers belong to each class?

Answer 37

The congruence classes modulo 6 are based on the remainder when dividing integers by 6. The classes are: Class 0 modulo 6: Numbers like ..., -12, -6, 0, 6, 12, 18, ... Class 1 modulo 6: Numbers like ..., -11, -5, 1, 7, 13, 19, ... Class 2 modulo 6: Numbers like ..., -10, -4, 2, 8, 14, 20, ... Class 3 modulo 6: Numbers like ..., -9, -3, 3, 9, 15, 21, ... Class 4 modulo 6: Numbers like ..., -8, -2, 4, 10, 16, 22, ... Class 5 modulo 6: Numbers like ..., -7, -1, 5, 11, 17, 23, ... Each class contains integers that leave the same remainder when divided by 6.