Problem 3
What are the domain and range of the addition function on the real numbers? On Multiplication? Subtraction? Division?
Problem 4
Show that the following sets are countably infinite:
(a) \(\mid q \in \mathbb{Q}: q>10\\}\)
(b) \(\left|q \in \mathbb{Q}: q^{2}
Problem 4
Let \(F: \mathbb{R} \rightarrow \mathbb{R}\) be defined as \(F(x)=2 x+8\). Let \(G: \mathbb{R} \rightarrow \mathbb{R}\) be defined as \(G(y)=\) \((y-8) / 2\). Prove that \(F \circ G=I d_{\mathrm{R}}\) and \(G \circ F=I d_{\mathrm{R}}\).
Problem 4
Prove that for any 44 people, at least four must be born in the same month.
Problem 4
Find the first six terms of the sequence with the elements defined as \(F(0)=5, F(1)=\) \(10,\) and \(F(n)=F(n-1)-2 F(n-2)\) for \(n \geq 2\).
Problem 5
(a) Prove that if \(X\) and \(Y\) are countable sets, so are \(X \cup Y, X \cap Y, X-Y\), and \(X \times Y\). (Caution: Countable means either finite or countably infinite, so there may be separate cases to consider.) (b) If \(X\) and \(Y\) are countably infinite, which of the following sets must be countably infinite: \(X \cup Y, X \cap Y, X-Y,\) and \(X \times Y ?\)
Problem 5
Find the first six terms of the sequence with the elements defined as \(F(0)=1, F(1)=\) 3, \(F(2)=5,\) and \(F(n)=3 F(n-1)+2 F(n-2)-3 F(n-3)\) for \(n \geq 3 .\)
Problem 5
Prove that in any class of 35 students, at least seven receive the same final grade, where the scale is \(\mathrm{A}-\mathrm{B}-\mathrm{C}-\mathrm{D}-\mathrm{F}\).
Problem 6
Find both a function defined by a formula and a recursively defined function for the following sequences: (a) \(1,3,5,7,9,11,13, \ldots\) (b) \(1,1,3,3,5,5,7,7, \ldots\) (c) \(0,2,4,6,8, \ldots\) (d) \(1,2,4,8,16, \ldots\)
Problem 6
Let \(X=\\{0,1,2\\} \subseteq \mathbb{R}\). List all eight strictly increasing sequences of elements of \(X\). The ordering is \(<\) on \(\mathbb{R}\). List all subsequences of the sequence \(x, y, z\).
