Enzyme, Pharmaceutical Aids, Miscellaneous Last Part of Chapter no 5th.pdf
5.3 Basic functions. A handout.
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