Definition 3.3.1 (Functions). Let \(X,Y\) be sets, and let \(P(x,y)\) be a property pertaining to an object \(x\in X\) and an object \(y\in Y\), such that for every \(x\in X\), there is exactly one \(y\in Y\) for which \(P(x,y)\) is true (this is sometimes known as the vertical line test). Then we define the function \(f:X\to Y\) defined by \(P\) on the domain \(X\) and range \(Y\) to be the object which, given any input \(x\in X\), assigns an output \(f(x)\in Y\), defined to be the unique object \(f(x)\) for which \(P(x,f(x))\) is true. Thus for any \(x\in X\) and \(y\in Y\), $$ y=f(x)\iff P(x,y)\text{ is true. } $$
Definition 3.3.1
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Exercise 3.3.1
\(\Rightarrow\) { \(f(x)\), \(g(x)\) and \(h(x)\) are objects by definition of function,
\(~~~\) transitive axiom for any three objects of the same type }\(\Rightarrow\) { \(f(x)\) and \(g(x)\) are objects by definition of function,
\(~~~\) the symmetry axiom for any two objects of the same type }\(\equiv\) { \(f(x)\) is an object by definition of function,
\(~~~\) reflexive axiom for any object }