91Ó°ÊÓ

Prove that any set \(S\) of three integers contains at least two integers whose sum is even. (Hint: Define a suitable function \(f: S \rightarrow\\{0,1\\}\) and use Exercise \(17 .\) )

Sums of the form \(S=\sum_{i=m+1}^{n}\left(a_{i}-a_{i-1}\right)\) are telescoping sums. Show that \(S=a_{n}-a_{m} .\)

Let \(A=\\{32,33, \ldots, 126\\} .\) Let \(f: A \rightarrow\) ASCII defined by \(f(n)=\) character with ordinal number \(n .\) Find \(f(n)\) for each value of \(n\). $$90$$

Let \(A\) be an \(m \times n\) matrix, \(B\) a \(p \times q\) matrix, and \(C\) an \(r \times s\) matrix. Under what conditions is each defined? Find the size of each when defined. (Note: \(A^{2}\) means \(A A .\) ) $$A B-A C$$

Let \(A=\left[\begin{array}{lll}{1} & {0} & {-1} \\ {0} & {2} & {3}\end{array}\right], B=\left[\begin{array}{rrr}{0} & {-2} & {5} \\ {0} & {0} & {1}\end{array}\right],\) and \(C=\left[\begin{array}{rrr}{-3} & {0} & {0} \\\ {0} & {1} & {2}\end{array}\right] .\) Find each. $$A B-A C$$

Determine if each function \(f: A \rightarrow B\) is bijective. $$f(x)=|x|, A=B=\mathbb{R}$$

Determine if each is true or false. Sums of the form \(S=\sum_{i=m+1}^{n}\left(a_{i}-a_{i-1}\right)\) are telescoping sums. Show that \(S=a_{n}-a_{m} . \quad i=m+1.\)

Determine if the given function is invertible. If it is not invertible, explain why. $$f: \mathbf{W} \rightarrow \mathbf{W} \text { defined by } f(n)=n(\bmod 5)$$

Using the pigeonhole principle, prove that the cardinality of a finite set is unique.

Let \(A=132,33, \ldots, 1261 .\) Let \(f : A \rightarrow\) ASCII defined by \(f(n)=\) character with ordinal number \(n .\) Find \(f(n)\) for each value of \(n .\) $$ 90 $$