I'm reading Terry Tao's book on Analysis 1.
I'm having trouble understanding how a function can be mapped as $\emptyset \rightarrow X$.
After stating the following map he said that all such maps for each set $X$ are equal, which I did not seem to understand.
Any help clearing the concept will be appreciated.
Best Answer
A map $f\colon A\to B$ is determined by its graph, which is the set subset of $A\times B$ consisting of the pairs $(a,f(a))$ for all $a\in A$. Indeed a subset $R$ of $A\times B$ is the graph of a map $A\to B$ if and only if for each $a\in A$ there is exactly one $b\in B$ such that $(a,b)\in R$. The corresponding map then has $f(a)=b$.
Now when $A=\varnothing$, the cartesian product $\varnothing\times B$ is empty, so the only possible graph of a function would be empty as well. And indeed this does satisfy all the properties: for each $a\in \varnothing$ there is a unique $b\in B$ such that $(a,b)\in\varnothing\times B = \varnothing$. This is satisfied since there is no $a\in\varnothing$ that could serve as a counter example.
We conclude that for any set $B$ there is only one map $\varnothing\to B$ and its graph is the empty set.