How-to-proof-f-X-Y-f-is-1-to-1-f-E-f-F-f-E-F-

Question Number 155586 by aaaspots last updated on 02/Oct/21

H o w t o p r o o f f : X \to Y

f i s 1 t o 1 \Leftarrow\Rightarrow f (E) ∖ f (F) = f (E ∖ F)

Answered by mindispower last updated on 04/Oct/21

b y D o u b l e i n c l u s i o n

l e t E, F b e e s u b s e t o F X

l e t y \in F (E ∖ F) \Rightarrow \exists! a \in E ∖ F s u c h f (a) = y

a \in E ∖ F \Rightarrow f (a) \in f (E), f (a) \notin f (F) s i n c e f i s i n j e c t i v e

\Rightarrow f (a) \in F (E) ∖ F (F) \Rightarrow f (E) ∖ f (F) \subseteq F (E ∖ F)

l e t z \in f (E ∖ F) \Rightarrow \exists! b \in E ∖ F ∣ f (b) = z

\Rightarrow f (b) \in f (E), f (b) \notin f (F) b e c a u s e i f f (b) \in f (F)

s u p p o s e f (b) \in f (F) \Rightarrow \exists a \in F f (a) = f (b) \Rightarrow a = b b y i n j e c t i v i t y

a = b, b \in F a b s u r d s i n c e b \in E ∖ F

\Rightarrow z \in f (E) ∖ f (F)

\Rightarrow f (E ∖ F) \subseteq f (E) ∖ f (F)

\Rightarrow f (E ∖ F) = f (E ∖ F)

Commented by aaaspots last updated on 04/Oct/21

h o w a b o u t 1 t o 1 \Leftarrow f (E ∖ F) = f (E) ∖ f (F)

Commented by mindispower last updated on 04/Oct/21

n o t t r u e i f w e t a c k F = \emptyset e m p t y s e t

b y d e f i n i t i o n f (\emptyset) = \emptyset

\Rightarrow \forall f \in f (X \to Y) \forall E \in P (X)

f i s 1 t o 1

Commented by aaaspots last updated on 07/Oct/21

a f t e r t h e t h i r d l i n e s I s t i l l u n d e r s t a n d