Problem 1
Prove that in any set of 27 words, at least two must begin with the same letter assuming at most a 26 -letter alphabet.
Problem 1
Show that if \(X \subseteq Y,|X| \leq|Y|\).
Problem 1
Which of the following are functions? If not, why not? (a) \(X\) is the set of students in the discrete mathematics class. For \(x \in X,\) define \(g(x)\) to be the youngest cousin of \(x\). (b) \(X\) is the set of senators serving in \(1998 .\) For \(x \in X,\) define \(g(x)\) to be the number of terms a senator has held. (c) For \(x \in \mathbb{R},\) define \(g(x)=|x /| x||\).
Problem 1
Let \(X=\\{1,2,3,4 \mid\) and \(Y=\\{5,6,7,8,9\\}\). Let \(F=I(1,5),(2,7),(4,9),(3,8)\\}\) Show that \(F\) is a function from \(X\) to \(Y\). Find \(F^{-1}\), and list its elements. Is \(F^{-1}\) a function? Why, or why not?
Problem 2
Prove that in any group of five integers, at least two have the same value under the \((\bmod 4)\) operation.
Problem 2
Let \(S=\\{(0,8),(1,10),(2,12),(3,14),(4,16),(5,18),(6,20),(7,22)]\). Is \(S\) a function? Why, or why not? Find \(S^{-1},\) and list its elements. Is \(S^{-1}\) a function? Why, or why not? Identify the domain of \(S^{-1}\)
Problem 2
Let \(X=\\{0,1, \ldots, 6,7\\}\) and \(Y=\\{8,10,12, \ldots, 20,22\\} .\) Define \(F: X \rightarrow Y\) as \(F(x)=2 x+8\). List the ordered pairs of the relation that define this function.
Problem 2
Prove that the sets \(\mathcal{X}=\\{2 n+1: n \in \mathbb{Z}\\}, \mathcal{Y}=\\{10 j: j \in \mathbb{Z}\\},\) and \(\mathcal{Z}=\\{3 n: n \in \mathbb{Z} \mid\) have the same cardinality.
Problem 3
Prove that in any class of more than 101 students, at least two must receive the same grade for an exam with grading scale of 0 to 100 .
Problem 3
In the first quadrant of the \(x-y\) plane, draw a path that passes exactly once through cach point with both coordinates being integers. Each stopping place on the path should only be one unit right, one unit up, one unit left, or one unit down from the previous stopping place. Start the path at (0,0) . Use the path to construct a bijection from \(\mathrm{N}\) to \(\mathrm{N} \times \mathrm{N}\).
