The document provides an introduction to set theory and the foundational concepts in the methodology and philosophy of mathematics and computer programming, focusing on basic functions. It defines several functions including identity, inclusion, constant, and characteristic functions, along with projections from Cartesian products. Additionally, it discusses specific propositions regarding functions between sets and their properties.

1. Introduction to set theory and to methodology and philosophy of
mathematics and computer programming
Basic functions
An overview
by Jan Plaza
c 2017 Jan Plaza
Use under the Creative Commons Attribution 4.0 International License
Version of November 6, 2017

2. Definition
1. The identity function on X , idX , is the function on X s.t.
idX(x)=x for every x ∈ X.
2. Let X ⊆ Y . The inclusion function from X to Y , X → Y , is idX.
3. f is a constant function on X if f is a function on X, and there is c such that
f(x) = c for every x ∈ X.
4. The empty function is the empty relation; ∅ : ∅ −→ X.
5. Let X ⊆ U. The characteristic function of X with respect to U is the
function from U to {0, 1} defined as follows:
x → 1 if x ∈ X,
x → 0 if x ∈ X.

3. True or false?
χA∩B(x) = χA(x) · χB(x),
True
True or false?
χA∪B(x) = χA(x) + χB(x)
False
Find a counter-example!

4. Definition
Let X1 and X2 be sets.
The 1-st projection from X1 × X2 is the function
π1 : X1 × X2 −→ X1 defined as π1 : x1, x2 → x1.
The 2-nd projection from X1 × X2 is the function
π2 : X1 × X2 −→ X2 defined as π2 : x1, x2 → x2.
Proposition
For every set Y and f1 : Y −→ X1 and f2 : Y −→ X2, there exists a unique function
f : Y −→ X1 × X2 such that π1 ◦ f = f1 and π2 ◦ f = f2.
Y
f1
zz
! f
f2
$$
X1 X1 × X2π1
oo
π2
// X2