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operationsres
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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 f : A \rightarrow B is some map. Then we define the graph of f by G(f) := \{(x,f(x)) \in A \times B : x \in A\}.(2) Other definition
A and B are sets and f : A \rightarrow B is some map. Further, define for every y \in B the corresponding intersection G_{fy} by G_{fy} := G(f) \cap \{(x,y) : x \in A\}.
(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 G_{fy} = G(f) based on my interpretation of (2), making G_{fy} superfluous.
(1) Definition (Graph of a map)
A and B are sets and f : A \rightarrow B is some map. Then we define the graph of f by G(f) := \{(x,f(x)) \in A \times B : x \in A\}.(2) Other definition
A and B are sets and f : A \rightarrow B is some map. Further, define for every y \in B the corresponding intersection G_{fy} by G_{fy} := G(f) \cap \{(x,y) : x \in A\}.
(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 G_{fy} = G(f) based on my interpretation of (2), making G_{fy} superfluous.