Chapter 3

Evaluate each sum. $$\sum_{k=-2}^{4} 3 k$$

Let \(f(x)=\lfloor x\rfloor\) and \( g(x)=\lceil x\rceil,\) where \(x \in \mathbb{R} .\) Compute each. $$(g \circ f)(-4.1)$$

There are six matching pairs of gloves. Show that any set of seven gloves will contain a matching pair.

Determine if each function is injective, where trunc( \(x\) ) denotes the integral part of the real number of \(x.\) $$f(x)=\lfloor x\rfloor, x \in \mathbb{R}$$

Let \(x=3.456\) and \(y=2.789 .\) Compute each. $$\lfloor x y\rfloor$$

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

Let \(f(x)=\lfloor x\rfloor\) and \(g(x)\) \(=\lceil x\rceil,\) where \(x \in \mathbb{R} .\) Compute each. $$(g \circ f)(-4.1)$$

Let \(x=3.456\) and \(y=2.789 .\) Compute each. $$\lfloor x y\rfloor$$

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

Evaluate each sum. $$\sum_{k=-2}^{3} 3\left(k^{2}\right)$$

Let \(f(x)=\lfloor x\rfloor\) and \( g(x)=\lceil x\rceil,\) where \(x \in \mathbb{R} .\) Compute each. $$(f \circ g)(-3.9)$$

Let \(x=3.456\) and \(y=2.789 .\) Compute each. $$\lfloor x\rfloor\lfloor y\rfloor$$

The sum of nine integers in the range \(1-25\) is \(83 .\) Show that one of them must be at least \(10 .\)

Determine if each function is injective, where trunc( \(x\) ) denotes the integral part of the real number of \(x.\) $$g(x)=\lceil x\rceil, x \in \mathbb{R}$$

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

Let \(f(x)=\lfloor x\rfloor\) and \(g(x)\) \(=\lceil x\rceil,\) where \(x \in \mathbb{R} .\) Compute each. $$(f \circ g)(-3.9)$$

Let \(x=3.456\) and \(y=2.789 .\) Compute each. $$\lfloor x\rfloor\lfloor y\rfloor$$

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

Evaluate each sum. $$\sum_{k=-1}^{3}(3 k)^{2}$$

Let \(f, g: \mathbf{W} \rightarrow \mathbf{W}\) defined by \(f(x)=x \bmod 5\) and \(g(x)=x \operatorname{div} 7 .\) Evaluate each. $$(g \circ f)(17)$$

Let \(x=3.456\) and \(y=2.789 .\) Compute each. $$\lfloor- x\rfloor$$

The total cost of 13 refrigerators at a department store is 12,305 dollar Show that one refrigerator must cost at least 947 dollar.

Determine if each function is injective, where trunc( \(x\) ) denotes the integral part of the real number of \(x.\) $$h(x)=\operatorname{trunc}(x), x \in \mathbb{R}$$

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

Let \(x=3.456\) and \(y=2.789 .\) Compute each. $$[-x]$$

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

Evaluate each sum. $$\sum_{k=1}^{5}(3-2 k) k$$

Let \(f, g: \mathbf{W} \rightarrow \mathbf{W}\) defined by \(f(x)=x \bmod 5\) and \(g(x)=x \operatorname{div} 7 .\) Evaluate each. $$(f \circ g)(23)$$

Mrs. Zee has 19 skirts and would like to arrange them in a chest that has four drawers. Show that one drawer must contain at least five skirts.

Determine if each function is injective, where trune(x) denotes the integral part of the real number of \(x .\) \(f : S \rightarrow\) W defined by \(f(A)=|A|,\) where \(S\) is the family of all finite sets.

\(f: S \rightarrow \mathbf{W}\) defined by \(f(\boldsymbol{A})=|\boldsymbol{A}|,\) where \(S\) is the family of all finite sets.

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

Let \(x=3.456\) and \(y=2.789 .\) Compute each. $$-\lfloor x\rfloor$$

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

Evaluate each sum. $$\sum_{j=-1}^{4}(j-2)^{2}$$

Let \(f, g: \mathbf{W} \rightarrow \mathbf{W}\) defined by \(f(x)=x \bmod 5\) and \(g(x)=x \operatorname{div} 7 .\) Evaluate each. $$(g \circ f)(97)$$

Let \(x=3.456\) and \(y=2.789 .\) Compute each. $$\lceil x+y\rceil$$

Show that the repeating decimal \(0 . a_{1} a_{2} \dots a_{i} b_{1} b_{2} \dots b_{j}\) is a rational number.

Let \(\Sigma=\\{0,1\\}\) . Let \(f : \Sigma^{*} \rightarrow\) W defined by \(f(x)=\|x\| .\) Evaluate \(f(x)\) for each value of \(x .\) 000101

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

Show that the repeating decimal 0. \(a_{1} a_{2} \ldots a_{i} \overline{b_{1} b_{2} \ldots b_{j}}\) is a rational number.

Let \(x=3.456\) and \(y=2.789 .\) Compute each. $$\lceil x+y\rceil$$

Let \(\Sigma=\\{0,1\\} .\) Let \(f: \Sigma^{*} \rightarrow \mathbf{W}\) defined by \(f(x)=\|x\| .\) Evaluate \(f(x)\) for each value of \(x\). $$000101$$

Determine if each function from \(\mathbb{R}\) to \(\mathbf{Z}\) is surjective. $$g(x)=\lfloor x\rfloor$$

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

Evaluate each sum. $$\sum_{i=0}^{5}(0.1)^{i}(0.9)^{5-i}$$

Let \(f, g: \mathbf{W} \rightarrow \mathbf{W}\) defined by \(f(x)=x \bmod 5\) and \(g(x)=x \operatorname{div} 7 .\) Evaluate each. $$(f \circ g)(78)$$

Let \(x=3.456\) and \(y=2.789 .\) Compute each. $$\lceil x\rceil+\lceil y\rceil$$

Let \(n \in \mathbb{N} .\) Suppose \(n\) elements are selected from the set \(\\{1,2, \ldots, 2 n\\}\) Find a pair of integers in which one is not a factor of another integer. Use the pigeonhole principle to prove the following.

Let \(\Sigma=\\{0,1\\}\) . Let \(f : \Sigma^{*} \rightarrow\) W defined by \(f(x)=\|x\| .\) Evaluate \(f(x)\) for each value of \(x .\) 1010100