The definition of a function given in my proofs class is:

$\Gamma f = \{(a,b)\in A\times\ B: b=f(a)\} = \{(a,\ f(a)): a\in A\} \subseteq A \times B$

And we are asked if the following are functions

$\{(2a,\ a^2): a\in \Bbb Z\} \subseteq \Bbb Z\ \times \Bbb Z$

$\{(a^2,\ a^3): a\in \Bbb Z\} \subseteq \Bbb Z\ \times \Bbb Z$

$\{(a^2,\ a^2): a\in \Bbb Z\} \subseteq \Bbb Z_{\ge0}\ \times \Bbb Z_{\ge0}$

Each answer is no they are not and I do not understand why. My thinking process is(using first one for an example) that the coordinates would be something like $(-4,4), (-2,1), (0,0), (2,1), (4,4)$. Which would form a parabola which is a function.

Is there something I'm missing or am I misinterpreting the definition?

  • $\begingroup$ That definition you say you were given is a rather poor one as it depends completely on what in the world $\;f\;$ is ...and if you already defined $\;;$ to be a function then you have one ugly circular definition. That looks close to be the definition of "graph" of a function... $\endgroup$
    – DonAntonio
    Commented Dec 14, 2017 at 6:49

1 Answer 1


For the first set, the image of odd number are not defined but odd number is included in the domain. Hence it is not a function. In particular, $f(1)$ is not defined.

For the other two sets, try to think of a number that is included in the domain but the image is not defined.


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