91Ó°ÊÓ

\((x, y) \in R\) if 3 divides \(x+2 y\)

Define a relation \(R\) on \(\mathbf{R}^{\mathbf{R}},\) the set of functions from \(\mathbf{R}\) to \(\mathbf{R}\), by \(f R g\) if \(f(0)=g(0)\). Prove that \(R\) is an equivalence relation on \(\mathbf{R}^{\mathbf{R}}\). Let \(f(x)=x\) for all \(x \in \mathbf{R}\). Describe \([f]\).

Each function in is one-to-one on the specified domain \(X .\) By letting \(Y=\) range off \(,\) we obtain a bijection from \(X\) to \(Y .\) Find each inverse function. $$ f(x)=4 x+2, X=\text { set of real numbers } $$

Not reflexive, symmetric, not antisymmetric, and transitive

If \(R\) is transitive, then \(R^{-1}\) is transitive.

If \(R\) is reflexive, then \(R^{-1}\) is reflexive.

For the sequence b defined by \(b_{n}=n(-1)^{n}, n \geq 1\). Find a formula for the sequence \(c\) defined by $$ c_{n}=\sum_{i=1}^{n} b_{i} $$

Show that if a single digit of an ISBN is changed, the check digit will change. Thus, any single-digit error will be detected.

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

Use the following definitions. Let \(X=\\{a, b, c\\}\) Define a function \(S\) from \(\mathcal{P}(X)\) to the set of bit strings of length 3 as follows. Let \(Y \subseteq X .\) If \(a \in Y,\) set \(s_{1}=1 ;\) if \(a \notin Y,\) set \(s_{1}=0 .\) If \(b \in Y,\) set \(s_{2}=1 ;\) if \(b \notin Y,\) set \(s_{2}=0 .\) If \(c \in Y,\) set \(s_{3}=1 ;\) if \(c \notin Y,\) set \(s_{3}=0 .\) Define \(S(Y)=s_{1} s_{2} s_{3}\). Prove that \(S\) is onto.