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Chapter 3: Problem 10
For the sequence t defined by \(t_{n}=2 n-1, \quad n \geq 1\). Find \(t_{3}\).
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For the sequence z defined by $$z_{n}=(2+n) 3^{n}, \quad n \geq 0$$. Find \(z_{3}\)
A binary operator \(f\) on a set \(X\) is commutative if \(f(x, y)=f(y, x)\) for all \(x, y \in X .\) In state whether the given function \(f\) is a binary operator on the set \(X .\) If \(f\) is not a binary operator, state why. State whether or not each binary operator is commutative. $$ f(x, y)=x / y, \quad X=\\{0,1,2, \ldots\\} $$
Using the sequences \(y\) and \(z\) defined by $$y_{n}=2^{n}-1, \quad z_{n}=n(n-1)$$. Find \(\left(\sum_{i=3}^{4} y_{i}\right)\left(\prod_{i=2}^{4} z_{i}\right)\)
In give an example of a unary operator \([\) different from \(f(x)=x\), for all \(x\) ] on the given set. $$ \\{\ldots,-2,-1,0,1,2, \ldots\\} $$
A binary operator \(f\) on a set \(X\) is commutative if \(f(x, y)=f(y, x)\) for all \(x, y \in X .\) In state whether the given function \(f\) is a binary operator on the set \(X .\) If \(f\) is not a binary operator, state why. State whether or not each binary operator is commutative. $$ f(x, y)=x^{2}+y^{2}-x y, \quad X=\\{1,2, \ldots\\} $$
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