91Ó°ÊÓ

Consider the function \(f: \mathbb{N} \rightarrow \mathbb{N}\) that gives the number of handshakes that take place in a room of \(n\) people assuming everyone shakes hands with everyone else. Give a recursive definition for this function.

Let \(f: X \rightarrow Y\) be a function, \(A \subseteq X\) and \(B \subseteq Y\). (a) Is \(f^{-1}(f(A))=A ?\) Always, sometimes, never? Explain. (b) Is \(f\left(f^{-1}(B)\right)=B ?\) Always, sometimes, never? Explain. (c) If one or both of the above do not always hold, is there something else you can say? Will equality always hold for particular types of functions? Is there some other relationship other than equality that would always hold? Explore.

Let \(f: X \rightarrow Y\) be a function and \(A, B \subseteq X\) be subsets of the domain. (a) Is \(f(A \cup B)=f(A) \cup f(B)\) ? Always, sometimes, or never? Explain. (b) Is \(f(A \cap B)=f(A) \cap f(B) ?\) Always, sometimes, or never? Explain.

Find all sets \(A, B,\) and \(C\) which satisfy the following. $$\begin{array}{l}A=\\{1,|B|,|C|\\} \\\B=\\{2,|A|,|C|\\} \\\C=\\{1,2,|A|,|B|\\} .\end{array}$$