Sets 2

Question 1

List (2)

Answer 1

1. Ordered sequence of elements.

2. Position and length are relevant characteristics

Question 2

Cartesian product (2)

Answer 2

1. A Cartesian product of two or more sets is a set of lists.
2. Each list will have n elements, which are the n of sets involved.

Question 3

Product Rule (1)

Answer 3

1. # (A * B) = #A * #B

Question 4

Binary relation (1)

Answer 4

1. Relation of type A * B or A^2

Question 5

Representations of relations (3)

Answer 5

1. Directed graphs
2. Matrices
3. Venn diagrams

Question 6

Inverse of a binary relation (2)

Answer 6

1. R^−1 := {⟨x,y⟩ : ⟨y,x⟩ ∈ R}

2. R⊆A×B ⇒ R^−1⊆B×A

Question 7

Inverse of a relation in representations (2)

Answer 7

1. In venn diagrams invert arrows

2. In Matrices transpose them.

Question 8

Composition (2)

Answer 8

1. R ◦ S := {⟨x, z⟩ : x S y and y R z for some y }

2. S⊆A×B,R⊆B×C ⇒ R◦S⊆A×C

Question 9

Properties of composition (2)

Answer 9

1. Composition is associative

2. (R ◦S)^−1 = S^−1 ◦R^−1

Question 10

Properties for relations in set A ()

Answer 10

1. Reflexivity: xRx
2. Transitivity: xRy∧yRz→xRz
3. Symmetry: x R y → y R x
4. Anti-symmetry: x R y → ¬y R x