91Ó°ÊÓ

For the sequence w defined by \(w_{n}=\frac{1}{n}-\frac{1}{n+1}, \quad n \geq 1\). Find \(\sum_{i=1}^{10} w_{i}\)

For the sequence w defined by \(w_{n}=\frac{1}{n}-\frac{1}{n+1}, \quad n \geq 1\). Find a formula for the sequence \(c\) defined by $$ c_{n}=\sum_{i=1}^{n} w_{i} $$

For the sequence w defined by \(w_{n}=\frac{1}{n}-\frac{1}{n+1}, \quad n \geq 1\). Find a formula for the sequence \(d\) defined by $$ d_{n}=\prod_{i=1}^{n} w_{i} $$

Let \(g\) be a function from \(X\) to \(Y\) and let \(f\) be a function from \(Y\) to Z. For each statement in if the statement is true, prove it; otherwise, give a counterexample. $$ \text { If } f \text { and } g \text { are onto, then } f \circ g \text { is onto. } $$

Let \(g\) be a function from \(X\) to \(Y\) and let \(f\) be a function from \(Y\) to Z. For each statement in if the statement is true, prove it; otherwise, give a counterexample. If \(f\) and \(g\) are one-to-one and onto, then \(f \circ g\) is one-to-one and onto.

For the sequence w defined by \(w_{n}=\frac{1}{n}-\frac{1}{n+1}, \quad n \geq 1\). Is \(w\) increasing?

For the sequence w defined by \(w_{n}=\frac{1}{n}-\frac{1}{n+1}, \quad n \geq 1\). Is \(w\) decreasing?

Let \(g\) be a function from \(X\) to \(Y\) and let \(f\) be a function from \(Y\) to Z. For each statement in if the statement is true, prove it; otherwise, give a counterexample. If \(f \circ g\) is one-to-one, then \(f\) is one-to-one.

Let \(g\) be a function from \(X\) to \(Y\) and let \(f\) be a function from \(Y\) to Z. For each statement in if the statement is true, prove it; otherwise, give a counterexample. If \(f \circ g\) is one-to-one, then \(g\) is one-to-one.

For the sequence w defined by \(w_{n}=\frac{1}{n}-\frac{1}{n+1}, \quad n \geq 1\). Is \(w\) nonincreasing?