91Ó°ÊÓ

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 \(n .\) $$ 38 $$

Prove the following alternate version of the generalized pigeonhole principle: Let \(f: X \rightarrow Y,\) where \(X\) and \(Y\) are finite sets, \(|X|>k \cdot|Y|\) and \(k \in \mathbb{N} .\) Then there is an element \(t \in Y\) such that \(f^{-1}(t)\) contains more than \(k\) elements.

Find the number of positive integers \(\leq 3076\) and divisible by: \(3,5,\) or 6

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 \(n .\) $$ 64 $$

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 \(n\). $$64$$

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. $$A(B+C)$$

Determine if the given function is invertible. If it is not invertible, explain why. \(f:\) ASCII \(\rightarrow \mathbf{W}\) defined by \(f(c)=\) ordinal number of the character \(c\).

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 \(17 .\) )

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 \(A A .\) ) $$A(B+C)$$

Determine if each is true or false. $$\sum_{i=m}^{n} x^{i}=\sum_{i=m}^{n} x^{n+m-i}$$