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Suppose I want to show that two functions f and g are equal. A way to prove this could be to prove the statement:

[tex] f(x) = n \Leftrightarrow g(x) = n[/tex]

Is it enough to show one side of the implication?

Prove the following statement:

[tex] f(x) = n \Rightarrow g(x) = n[/tex]

and reason as follows, suppose [tex] f(x) \neq n[/tex],

[tex] \Rightarrow \exists m\neq n: f(x)=m[/tex]

[tex]\Rightarrow g(x)=m\neq n[/tex]

Which would mean that I have shown the converse implication, and thus I have equivalence.

[tex] f(x) = n \Leftrightarrow g(x) = n[/tex]

Is it enough to show one side of the implication?

Prove the following statement:

[tex] f(x) = n \Rightarrow g(x) = n[/tex]

and reason as follows, suppose [tex] f(x) \neq n[/tex],

[tex] \Rightarrow \exists m\neq n: f(x)=m[/tex]

[tex]\Rightarrow g(x)=m\neq n[/tex]

Which would mean that I have shown the converse implication, and thus I have equivalence.

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