5.4: Section 4-
- Page ID
- 33717
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)A. Prove the following statements with contrapositive proof. (In each case, think about how a direct proof would work. In most cases contrapositive is easier.)
Exercise \(\PageIndex{1}\)
Suppose \(n \in \mathbb{Z}\). If \(n^2\) is even, then n is even.
Exercise \(\PageIndex{2}\)
Suppose \(n \in \mathbb{Z}\). If \(n^2\) is odd, then n is odd.
Exercise \(\PageIndex{3}\)
Suppose \(a, b \in \mathbb{Z}\). If \(a^{2}(b^2-2b)\) is odd, then a and b are odd.
Exercise \(\PageIndex{4}\)
Suppose \(a, b, c \in \mathbb{Z}\). If a does not divide bc, then a does not divide b.
Exercise \(\PageIndex{5}\)
Suppose \(x \in \mathbb{R}\). If \(x^2+5x < 0\) then \(x < 0\).
Exercise \(\PageIndex{6}\)
Suppose \(x \in \mathbb{R}\). If \(x^3-x > 0\) then \(x > -1\).
Exercise \(\PageIndex{7}\)
Suppose \(a, b \in \mathbb{Z}\). If both ab and \(a+b\) are even, then both a and b are even.
Exercise \(\PageIndex{8}\)
Suppose \(x \in \mathbb{R}\). If \(x^5-4x^4+3x^3-x^2+3x-4 \ge 0\), then \(x \ge 0\).
Exercise \(\PageIndex{9}\)
Suppose \(n \in \mathbb{Z}\). If \(3�� \nmid n^2\), then \(3�� \nmid n\).
Exercise \(\PageIndex{10}\)
Suppose \(x, y, z \in \mathbb{Z}\) and \(x \ne 0\). If \(x \nmid ��yz\), then \(x \nmid ��y\) and \(x \nmid ��z\).
Exercise \(\PageIndex{11}\)
Suppose \(x, y \in \mathbb{Z}\). If \(x^{2}(y+3)\) is even, then x is even or y is odd.
Exercise \(\PageIndex{12}\)
Suppose \(a \in \mathbb{Z}\). If \(a^2\) is not divisible by 4, then a is odd.
Exercise \(\PageIndex{13}\)
Suppose \(x \in \mathbb{R}\). If \(x^5+7x^3+5x \ge x^4+x^2+8\), then \(x \ge 0\).
B. Prove the following statements using either direct or contrapositive proof.
Exercise \(\PageIndex{14}\)
If \(a, b \in \mathbb{Z}\) and a and b have the same parity, then \(3a+7\) and \(7b-4\) do not.
Exercise \(\PageIndex{15}\)
Suppose \(x \in \mathbb{Z}\). If \(x^3-1\) is even, then x is odd.
Exercise \(\PageIndex{16}\)
Suppose \(x, y \in \mathbb{Z}\). If \(x+y\) is even, then x and y have the same parity.
Exercise \(\PageIndex{17}\)
If n is odd, then \(8|(n^2-1)\).
Exercise \(\PageIndex{18}\)
If \(a, b \in \mathbb{Z}\), then \((a+b)^{3} \equiv a^3+b^3 \pmod{3}\).
Exercise \(\PageIndex{19}\)
Let \(a, b, c \in \mathbb{Z}\) and \(n \in \mathbb{N}\). If \(a \equiv b \pmod{n}\) and \(a \equiv c \pmod{n}\), then \(c \equiv b \pmod{n}\).
Exercise \(\PageIndex{20}\)
If \(a \in \mathbb{Z}\) and \(a \equiv 1 \pmod{5}\), then \(a^2 \equiv 1 \pmod{5}\).
Exercise \(\PageIndex{21}\)
Let \(a, b \in \mathbb{Z}\) and \(n \in \mathbb{N}\). If \(a \equiv b \pmod{n}\), then \(a^3 \equiv b^3 \pmod{n}\).
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Let a∈Z, n∈N. If a has remainder r when divided by n, then a≡r (mod n).
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Leta,b∈Zandn∈N.Ifa≡b(modn),thena2≡ab(modn).
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If a≡b (mod n) and c≡d (mod n), then ac≡bd (mod n).
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Letn∈N.If2n−1isprime,thennisprime.
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Ifn=2k−1fork∈N,theneveryentryinRownofPascal’sTriangleisodd.
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If a≡0 (mod 4) or a≡1 (mod 4), then ��a2�� is even.
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Ifn∈Z,then4��(n2−3).
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Ifintegersaandbarenotbothzero,thengcd(a,b)=gcd(a−b,b).
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Ifa≡b(modn),thengcd(a,n)=gcd(b,n).
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Suppose the division algorithm applied to a and b yields a = qb + r. Prove gcd(a, b) = gcd(r, b).
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If a ≡ b (mod n), then a and b have the same remainder when divided by n