Logic as a tool a guide to formal logical reasoning ( PDFDrive ) 261

k Applications: Mathematical Proofs and Automated Reasoning 237 Whenever we refer to a compositiongf of two mappingsf andg, we will assume that rngf⊆ dom g.. Proposition 193 Composition

k

Applications: Mathematical Proofs and Automated Reasoning 237

Whenever we refer to a compositiongf of two mappingsf andg, we will assume that rng(f)⊆ dom( g)

Proposition 193 Composition of mappings is associative, that is, f(gh) = (f g)h , when-ever either of these is defined.

Proposition 194 Let f :A → B , g:B → C be any mappings Then the following hold.

(a) If f and g are injective then gf is also injective.

(b) If f and g are surjective then gf is also surjective.

(c) In particular, the composition of bijections is a bijection.

(d) If gf is injective then f is injective.

(e) If gf is surjective then g is surjective.

Proposition 195 If f :A → B , g:B → C are bijective mappings then gf has an inverse, such that ( gf)−1=f −1 g −1

Proposition 196

1 A mapping f is injective iff for every two mappings g1and g2withdom(g1) =dom(g2)

andcod(g1) =dom(f) =cod(g2), the following left cancellation property holds:

if f g1=f g2then g1 =g2.

2 A mapping f is surjective iff for every two mappings g1 and g2 with dom(g1) = dom(g2) =cod(f ), the following right cancellation property holds:

if g1f =g2f then g1 =g2. 5.2.4 Binary relations and operations on them

We have already discussed relations (predicates) of any number of arguments Being sub-sets of a given universal set, sub-sets can be regarded as unary relations Here we focus on

binary relations and operations on them.

First, let us summarize the basic terminology Given setsAandB, a binary relation

between A and B is any subset of A × B In particular, a binary relation on a set

Ais any subset ofA2 =A × A Given a binary relationR ⊆ A × B, if (a, b)∈ R, we sometimes also writeaRband say thataisR-related tob

Given setsAandBand a binary relationR ⊆ A × B:

• the domain of Ris the setdom(R) ={ a ∈ A | ∃ b ∈ B(aRb)}; and

• the range of Ris the setrng(R) ={ b ∈ B | ∃ a ∈ A(aRb)}.

More generally, given subsetsX ⊆ AandY ⊆ B, we define

• the image of XunderR:R[X] ={ b ∈ B | ∃ x ∈ X(xRb)}; and

• the inverse image of Y underR:R −1[Y] ={ a ∈ A | ∃ y ∈ Y(aRy)}.

Notice thatdom(R) =R −1[B] andrng(R) =R[A]