Problem set 1: Mathematical Preliminaries
1. Problem set 1: Mathematical Preliminaries#
Let \(A\), \(B\) and \(C\) be sets. Show that \(A \cup (B \cap C ) = (A \cup B ) \cap (A \cup C)\), and \(A \cap (B \cup C) = (A \cap B ) \cup (A \cap C )\).
Let \(A\) and \(B\) be sets. Show that \(A-B = \emptyset\) implies \(A \subset B\).
Show that for any sets \(A\), \(B\), \(C\), \(D\), \((A \bigotimes B) \cap (C \bigotimes D) = (A \cap C) \bigotimes (B \cap D)\).
Show that for any function \(f\) with domain \(\mathcal{X}\), if \(A, B \subset \mathcal{X}\), then \(f(A \cup B ) = fA \cup fB\). Give an example that shows \(f(A \cap B)\) is not necessarily equal to \(fA \cap fB\).
Let \(f\) be a function with co-domain \(\mathcal{Y}\), and \(A, B \subset \mathcal{Y}\). Does \(f^{-1} (A \cap B) = f^{-1}A \cap f^{-1}B\)? Does \(f^{-1} (A \cup B ) = f^{-1}A \cup f^{-1} B\)?
Let \(f\) have domain \(\mathcal{X}\) and co-domain \(\mathcal{Y}\), and suppose that \(A \subset \mathcal{X}\) and \(B \subset \mathcal{Y}\). Does \(f^{-1}(f(A)) = A\)? Does \(f(f^{-1}B) = B\)?
Let \(\mathcal{G}\) be a group with identity \(e\). Show that \(ae = (a^{-1})^{-1} = a\). (That is, show that \(e\) is not only the identity from the left, it is also the identity from the right, and that if \(a^{-1}a = e\), then \(aa^{-1} = e\).)
Let \(a, b, c, d \in F\), where \(F\) is a field. Show that if \(b, d \ne 0\), then \(a/b+c/d = (ad+bc)/bd\).
Show that \(A= \{0,1,2, \cdots, p-1 \}\) with \(p\) prime is a field, if addition and multiplication are defined modulo \(p\). What breaks down if \(p\) is not prime? For \(p=7\), show that the multiplicative inverse of 2 is 4.