Chapter 3

Expand each. $$\sum_{1 \leq i \leq j<3}\left(a_{i}+a_{j}\right)$$

(Easter Sunday) The date for Easter Sunday in any year \(y\) can be computed as follows. Let \(a=y \bmod 19, b=y \bmod 4, c=y \bmod 7, d=(19 a+24)\) \(\bmod 30, e=(2 b+4 c+6 d+5) \bmod 7,\) and \(r=(22+d+e) .\) If \(r \leq 31,\) then Easter Sunday is March \(r ;\) otherwise, it is April \([r(\bmod 31)] .\) Compute the date for Easter Sunday in each year. $$3000$$

The inverse of \(f\) is unique.

Let \(f: X \rightarrow Y\) and \(g: Y \rightarrow Z\) be invertible functions. Prove each. The inverse of \(f\) is unique.

Prove each, where \(x \in \mathbb{R}\) and \(n \in \mathbf{Z}.\) \(\left\lfloor\frac{n}{2}\right\rfloor=\frac{n-1}{2}\) if \(n\) is odd.

Let \(M\) denote the set of \(2 \times 2\) matrices over \(\mathbf{w} .\) Let \(f : N \rightarrow M\) defined by \(f(n)=\left[\begin{array}{ll}{1} & {1} \\ {1} & {0}\end{array}\right]^{n} .\) Compute \(f(n)\) for each value of \(n.\) $$2$$

Let \(f: A \rightarrow B, g: B \rightarrow C,\) and \(h: C \rightarrow D .\) Prove that \(h \circ(g \circ f)=\) \((h \circ g) \circ f\) (associative property). [Hint: Verify that \((h \circ(g \circ f))(x)=((h \circ g) \circ f)(x) \text { for every } x \text { in } A .]\)

Prove each, where \(x \in \mathbb{R}\) and \(n \in \mathbf{Z}\) $$\left\lfloor\frac{n}{2}\right\rfloor=\frac{n-1}{2} \text { if } n \text { is odd }$$

Prove each, where \(x \in \mathbb{R}\) and \(n \in \mathbf{Z}.\) \(\left\lceil\frac{n}{2}\right\rceil=\frac{n+1}{2}\) if \(n\) is odd

Let \(M\) denote the set of \(2 \times 2\) matrices over \(\mathbf{w} .\) Let \(f : N \rightarrow M\) defined by \(f(n)=\left[\begin{array}{ll}{1} & {1} \\ {1} & {0}\end{array}\right]^{n} .\) Compute \(f(n)\) for each value of \(n.\) $$3$$

Let \(f\) and \(g\) denote the functions defined by the if-then-else statements in Example \(3.31 .\) Show that \(g \circ f\) is defined as given in the example. (Hint: Consider the cases \(x \leq 4\) and \(x>4,\) and then two subcases in each case.)

Prove each, where \(x \in \mathbb{R}\) and \(n \in \mathbf{Z}\) $$\left\lceil\frac{n}{2}\right\rceil=\frac{n+1}{2} \text { if } n \text { is odd }$$

Prove each, where \(x \in \mathbb{R}\) and \(n \in \mathbf{Z}.\) \(\left\lfloor\frac{n^{2}}{4}\right\rfloor=\frac{n^{2}-1}{4}\) if \(n\) is odd

Let \(M\) denote the set of \(2 \times 2\) matrices over \(\mathbf{w} .\) Let \(f : N \rightarrow M\) defined by \(f(n)=\left[\begin{array}{ll}{1} & {1} \\ {1} & {0}\end{array}\right]^{n} .\) Compute \(f(n)\) for each value of \(n.\) $$4$$

Prove each, where \(X \sim Y\) implies set \(X\) is equivalent to set \(Y\). If \(A \sim B,\) then \(B \sim A\) (symmetric property).

Prove each, where \(x \in \mathbb{R}\) and \(n \in \mathbf{Z}\) $$\left|\frac{n^{2}}{4}\right|=\frac{n^{2}-1}{4} \text { if } n \text { is odd }$$

Prove each, where \(x \in \mathbb{R}\) and \(n \in \mathbf{Z}.\) \(\left[\frac{n^{2}}{4}\right]=\frac{n^{2}+3}{4}\) if \(n\) is odd

Let \(M\) denote the set of \(2 \times 2\) matrices over \(\mathbf{w} .\) Let \(f : N \rightarrow M\) defined by \(f(n)=\left[\begin{array}{ll}{1} & {1} \\ {1} & {0}\end{array}\right]^{n} .\) Compute \(f(n)\) for each value of \(n.\) $$5$$

Prove each, where \(X \sim Y\) implies set \(X\) is equivalent to set \(Y\). If \(A \sim B\) and \(B \sim C,\) then \(A \sim C\) (transitive property).

Prove each, where \(x \in \mathbb{R}\) and \(n \in \mathbf{Z}\) $$\left[\frac{n^{2}}{4}\right]=\frac{n^{2}+3}{4} \text { if } n \text { is odd }$$

Prove each, where \(x \in \mathbb{R}\) and \(n \in \mathbf{Z}.\) \(\left\lfloor\frac{n}{2}\right\rfloor+\left\lceil\frac{n}{2}\right\rceil= n\)

Prove each. The inverse of a square matrix \(A\) is unique. (Hint: Assume \(A\) has two inverses \(B\) and \(C\) . Show that \(B=C\) . \()\)

Let \(f: X \rightarrow Y\) be bijective. Let \(S\) and \(T\) be subsets of \(Y .\) Prove each. $$f^{-1}(S \cup T)=f^{-1}(S) \cup f^{-1}(T)$$

Prove each, where \(x \in \mathbb{R}\) and \(n \in \mathbf{Z}\) $$\left\lfloor\frac{n}{2}\right\rfloor+\left\lceil\frac{n}{2}\right\rceil=n$$

The inverse of a square matrix \(A\) is unique. (Hint: Assume \(A\) has two inverses \(B\) and \(C .\) Show that \(B=C .\) )

Prove each. If \(A\) is an invertible matrix, then \(\left(A^{-1}\right)^{-1}=A.\)

Let \(f: X \rightarrow Y\) be bijective. Let \(S\) and \(T\) be subsets of \(Y .\) Prove each. $$f^{-1}(S \cap T)=f^{-1}(S) \cap f^{-1}(T)$$

Prove each, where \(x \in \mathbb{R}\) and \(n \in \mathbf{Z}\) $$\lceil x\rceil=\lfloor x\rfloor+1 \quad(x \notin \mathbf{Z})$$

If \(A\) is an invertible matrix, then \(\left(A^{-1}\right)^{-1}=A\).

Prove each, where \(x \in \mathbb{R}\) and \(n \in \mathbf{Z}.\) $$\lceil x\rceil=-\lfloor- x\rfloor$$

Prove each. If \(A\) and \(B\) are two invertible matrices of order \(n,\) then \((A B)^{-1}=B^{-1} A^{-1} .\)

Prove each, where \(x \in \mathbb{R}\) and \(n \in \mathbf{Z}\) $$\lceil x\rceil=-\lfloor-x\rfloor$$

If \(A\) and \(B\) are two invertible matrices of order \(n,\) then \((A B)^{-1}=\) \(B^{-1} A^{-1}\).

Prove each, where \(x \in \mathbb{R}\) and \(n \in \mathbf{Z}\) $$\lceil x+n\rceil=\lceil x\rceil+n$$

Write an algorithm to compute the sum of the matrices \(A=\left(a_{i j}\right)_{m \times n}\) and \(B=\left(b_{i j}\right)_{m \times n}\).

Let \(A\) and \(B\) be any two sets, and \(U\) the universe. Let \(f_{S}\) denote the characteristic function of a subset \(S\) of \(U\) and \(x\) an arbitrary element in \(U .\) Prove each. $$f_{A \cup B}(x)=f_{A}(x)+f_{B}(x)-f_{A \cap B}(x)$$

Let \(A\) and \(B\) be any two sets, and \(U\) the universe. Let \(f_{S}\) denote the characteristic function of a subset \(S\) of \(U\) and \(x\) an arbitrary element in \(U .\) Prove each. $$f_{A^{\prime}}(x)=1-f_{A}(x)$$

Let \(A\) and \(B\) be any two sets, and \(U\) the universe. Let \(f_{S}\) denote the characteristic function of a subset \(S\) of \(U\) and \(x\) an arbitrary element in \(U .\) Prove each. $$f_{A \oplus B}(x)=f_{A}(x)+f_{B}(x)-2 f_{A \cap B}(x)$$

Let \(x, y \in \mathbb{R} .\) Let \(\max \\{x, y\\}\) denote the maximum of \(x\) and \(y,\) and \(\min \\{x, y\\}\) denote the minimum of \(x\) and \(y .\) Prove each. $$\max \\{x, y\\}+\min \\{x, y\\}=x+y$$

Let \(x, y \in \mathbb{R} .\) Let \(\max \\{x, y\\}\) denote the maximum of \(x\) and \(y,\) and \(\min \\{x, y\\}\) denote the minimum of \(x\) and \(y .\) Prove each. $$\max \\{x, y\\}-\min \\{x, y\\}=|x-y|$$