A. Sets, Relations and Functions

Question 1

What are members?

Answer 1

The elements of a set

Question 2

How to indicate a is an element of a set A?

Answer 2

a ∈ A

Question 3

When are 2 sets identical?

Answer 3

(1) if every element a of set A is also an element of set B
(2) if every element b of set B is also an element of set A

Question 4

Explain the difference between ∅ and {∅}

Answer 4

Set {∅} is a set with a member ∅, but ∅ is a set containing no member

Question 5

What is the cartesian product of A x B?

Answer 5

A × B = {(a, 1),(a, 2), . . . ,(b, 1),(b, 2), . . . ,(h, 1),(h, 2), . . . ,(h, 8)}

Question 6

What does Rˇ ⊆ R describe?

Answer 6

symmetry

Question 7

What does A² - I say?

Answer 7

All pars (x, y) ∈ A² with x /= y

Question 8

Describe asymmetric

Answer 8

∀x∀y(Rxy → ¬Ryx)

Question 9

Describe irreflexivity

Answer 9

∀x¬Rxx

Question 10

What is a serial relation?

Answer 10

∀x∃yRxy there is always a relation from x to y

Question 11

What is a function and what is it’s property?

Answer 11

Relations with the property that for any (a,b) and (a,c) it has to hold that b and c are equal.
Thus a function is a relation between A en B such that for each a ∈ A there is one and only one associated b ∈ B

Question 12

Give the function for mapping Kelvin to Celsius

Answer 12

x |→ x − 273.15

Question 13

What is a characteristic function?

Answer 13

The characteristic function of a subset A of some universe U is a function that maps all members of A to the truth-value True and all elements of U that are not members of A to False.

Question 14

What is the characterstic function of ≤ ?

Answer 14

f : N2 → {True, False}.
For every pair (m, n) ∈ N
2
, if m ≤ n then f((m, n)) = True, otherwise, f((m, n)) = false

Question 15

What is the cartesian product of:
A x B where
A = {(1,2),(3,4)}
B {(5,6)}

Answer 15

A x B = {((1,2),(5,6)), ((3,4),(5,6))}