Sets 3

Question 1

Partial order (4)

Answer 1

1. Relation on set V of the type V x V
2. Reflexive
3. Transitive
4. Anti-symmetric

Question 2

Power set (1)

Answer 2

1. Power set of V is the set of all subsets of V

Question 3

Comparable (1)

Answer 3

1. If the partial order R is: x R y or y R x

Question 4

Linear or Total order (1)

Answer 4

1. If all x,y ∈ V are comparable.

Question 5

Strict partial order (2)

Answer 5

1. A strict partial order is irreflexive

2. x < y ⇐⇒ x ≤ y and x ≠ y

Question 6

Hasse diagrams (1)

Answer 6

1. Directed graphs which omit reflexive and transitivity implied arrows.

Question 7

Lexicographic order (2)

Answer 7

1. On a set A x B, A takes priority over B

2. ≤ ⇐⇒ (a1 <a></a>

Question 8

Cartesian Order (4)

Answer 8

1. On a set A x B, A and B are treated equally
2. ≤ ⇐⇒ a1 ≤A a2 and b1 ≤B b2
3. Cartesian orders are partial orders
4. They are not total.

Question 9

Maximum (1)

Answer 9

1. m is the maximum if ∀a ∈ A : a ≤ m
2. Every maximum of A is a maximal element of A
3. If A has a maximum it is the only maximal element.

Question 10

Minimum (1)

Answer 10

1. m is the minimum if ∀a ∈ A : m ≤ a

2. Every minimum of A is a minimum element of A

Question 11

Maximal element (1)

Answer 11

1. m is a maximal element if ∀a ∈ A : (m ≤ a → m = a)

Question 12

Minimal element (1)

Answer 12

1. m is a minimal element if ∀a ∈ A : (a ≤ m → a = m)