Question

define an invertible mapping. prove that the inverse of an invertible mapping is invertible

Answer 1

Let f:S→T be a mapping.

Let f−1⊆T×S be the inverse of f:

f−1:={(t,s):f(s)=t}

Let f−1 itself be a mapping:

∀y∈T:(y,x1)∈f−1∧(y,x2)∈f−1⟹x1=x2

and

∀y∈T:∃x∈S:(y,x)∈f

Then f−1 is called the inverse mapping of f.

Definition 2

Let f:S→T and g:T→S be mappings.

Let:

g∘f=IS

f∘g=IT

where:

g∘f and f∘g denotes the composition of f with g in either order

IS and IT denote the identity mappings on S and T respectively.

That is, f and g are both left inverse mappings and right inverse mappings of each other.

Then:

g is the inverse (mapping) of f

f is the inverse (mapping) of g.

hope this helps you..

Answer 2

Answer:

Step-by-step explanation: