operationsres
- 99
- 0
A first year real analysis textbook presents the following two definitions (where the second builds off the first.
(1) Definition (Graph of a map)
A and B are sets and [itex]f : A \rightarrow B[/itex] is some map. Then we define the graph of [itex]f[/itex] by [tex]G(f) := \{(x,f(x)) \in A \times B : x \in A\}[/tex].(2) Other definition
A and B are sets and [itex]f : A \rightarrow B[/itex] is some map. Further, define for every [itex]y \in B[/itex] the corresponding intersection [itex]G_{fy}[/itex] by [tex]G_{fy} := G(f) \cap \{(x,y) : x \in A\}[/tex].
(This then proceeds into a theorem about bijections).
1. The problem I'm having
I completely understand (1) and all notation employed in both (1) and (2). However, I don't understand what (2) is trying to communicate ... It seems to me that [itex]G_{fy} = G(f)[/itex] based on my interpretation of (2), making [itex]G_{fy}[/itex] superfluous.
(1) Definition (Graph of a map)
A and B are sets and [itex]f : A \rightarrow B[/itex] is some map. Then we define the graph of [itex]f[/itex] by [tex]G(f) := \{(x,f(x)) \in A \times B : x \in A\}[/tex].(2) Other definition
A and B are sets and [itex]f : A \rightarrow B[/itex] is some map. Further, define for every [itex]y \in B[/itex] the corresponding intersection [itex]G_{fy}[/itex] by [tex]G_{fy} := G(f) \cap \{(x,y) : x \in A\}[/tex].
(This then proceeds into a theorem about bijections).
1. The problem I'm having
I completely understand (1) and all notation employed in both (1) and (2). However, I don't understand what (2) is trying to communicate ... It seems to me that [itex]G_{fy} = G(f)[/itex] based on my interpretation of (2), making [itex]G_{fy}[/itex] superfluous.