1.E: Exercises
- Page ID
- 7421
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Exercise \(\PageIndex{1}\): Binary operations
Evaluate the following:
- \(3 \oplus 4\)
- \(3 \ominus 4\)
- \(3\oslash 4\)
- \(3 \otimes 4\)
Exercise \(\PageIndex{2}\): Ominus
For \(a, b \in \mathbb{Z},\) define an operation \( \ominus\), by \( a \ominus b= ab+a-b.\) Determine whether \( \ominus\) on \(\mathbb{Z}\),
- is closed,
- Answer
-
Proof:
Let \(a, b \in \mathbb{Z}.\)
Then consider\( a \ominus b= ab+a-b.\)
Since ·, + and - are closed on \(\mathbb{Z}\), \(ab, (ab)+a, \) and \((ab+a)-b \) are in \(\mathbb{Z}\).
Hence \(a \ominus b \in \mathbb{Z}.\)
Thus, the binary operation is closed on \(\mathbb{Z}\). ⬜
2. is commutative,
- Answer
-
Counterexample:
Choose \(a=3\) and \( b=4.\)
Then consider \(3 \ominus 4=(3)(4)+3-4 =12+3-4 =15-4=11\).
Now consider \( 4 \ominus 3=(4)(3)+4-3=12+4-3 =16-3 =13.\)
Since \(11 \ne 13,\) the binary operation \( \ominus \) is not commutative on \(\mathbb{Z}\). ⬜
3. is associative,
- Answer
-
Counterexample:
Choose \( a=3, b=4, c=2.\)
Then consider \( (3 \ominus 4) \ominus 2=[(3)(4)+3-4] \ominus 2 =11 \ominus 2 =(11)(2)+11-2 =22+11-2 =31. \)
Now consider, \( 3 \ominus (4 \ominus 2)=3 \ominus [(4)(2)+4-2]=3 \ominus 10 =(3)(10)+3-10 =30+3-10 =23. \)
Since \(31 \ne 23, \) the binary operation is not associative on \(\mathbb{Z}\). ⬜
4. has an identity.
- Answer
-
Let \(e\) be the identity on \(\mathbb{Z}\). Then \(a \ominus e=e \ominus a=a, a∈\mathbb{Z}\).
Now consider, \( a \ominus e=e \ominus a.\)
\(ae+a-e=ea+e-a.\)
\( ae-ea+2a=2e.\)
\(ae=ea \) since · is commutative on \(\mathbb{Z}\).
Thus, \(e=a.\)
Now consider \( ae=ae+a-e=a.\)
If\( e=a,\) then \(a^2=a, a∈ \mathbb{Z}\).
This is a contradiction.
Thus, \( (\mathbb{Z}, \ominus ) \) has no identity.
Exercise \(\PageIndex{3}\): Oplus
For \(a, b \in \mathbb{Z},\) define an operation \( \oplus \), by \( a \oplus b= ab+a+b.\) Determine whether \( \oplus\) on \(\mathbb{Z}\),
- is closed,
- is commutative,
- is associative, and
- has an identity.
What happens to the result for the above questions, if we change the set to \(\mathbb{Z} \setminus \{-1\}\)?
Exercise \(\PageIndex{4}\): Oslash
For \(a, b \in \mathbb{Z},\) define an operation \( \oslash\), by \( a \oslash b= (a+b)(a-b).\) Determine whether \( \oslash \) on \(\mathbb{Z}\),
- is closed,
- is commutative,
- is associative, and
- has an identity.
Exercise \(\PageIndex{5}\): Otimes
For \(a, b \in \mathbb{Z},\) define an operation \( \otimes \), by \( a \otimes b= (a+b)(a+b).\) Determine whether \( \otimes \) on \(\mathbb{Z}\),
- is closed,
- Answer
-
Proof:
Let \( a, b \in \mathbb{Z}\). Then consider, \( a \otimes b= (a+b)(a+b).\)
since · is commutative on \(\mathbb{Z}\), \((a+b)(a+b)=a^2+2ab+b^2\). Since ·, + are closed on \(\mathbb{Z}\), \(a^2, 2ab, b^2, a^2+2ab, and (a^2+2ab)+b^ 2 \in \mathbb{Z}\).
Hence, \(a^2+2ab+b^ 2 \in \mathbb{Z}\). Thus \( a \otimes b \in \mathbb{Z}\).
Therefore, the binary operation \( \otimes \) is closed on \(\mathbb{Z}\).⬜
2. is commutative,
- Answer
-
Proof:
Let \( a, b \in \mathbb{Z}\). Then consider, \( a \otimes b= (a+b)(a+b) =a^2+2ab+b^2= b^2+2ba+a^2 = (b+a)(b+a)= b \otimes a .\)
Therefore, the binary operation \( \otimes \) is commutative on \(\mathbb{Z}\).⬜
3. is associative,
- Answer
-
Counterexample:
Choose \( a=2, b=3, c=4.\)
Then consider, \( ( a \otimes b ) \otimes c=((2 + 3)(2 + 3)) \otimes 4= 25 \otimes 4= (25 + 4)(25 + 4)=841 \).
Now consider \( a \otimes (b \otimes c)=2 \otimes (3 + 4)(3+ 4)= 2 \otimes 49 =(2 +49)(2 + 49) =2601.\)
Since \( 841 \ne 2601\), the binary operation \( \otimes\) is not associative on \(\mathbb{Z}\). ⬜
4. has an identity.
- Answer
-
Let \(e\) be the identity on \( ( \mathbb{Z}, \otimes \).
Then consider, \(a \otimes e=e \otimes a=a, \forall \mathbb{Z}.\)
Now, \(a = e \otimes a = (e+a)(e+a)= e(e+a)+a(e+a)\), since · is associative on \(\mathbb{Z}.\)
\(a = e^2+ea+ae+a^2=e^2+2ae+a^2. \) Then choose \(a=0.\)
Thus, \(e^2 = 0 \), which implies \( e = 0.\)
Hence \(a^ 2 = a, \forall \mathbb{Z}.\) This is a contradiction.
Thus,\( ( \mathbb{Z}, \otimes \) has no identity. ⬜
Exercise \(\PageIndex{6}\): Max
For \(a, b \in \mathbb{Z},\) define an operation \( \land \), by \( a\land b= max \{a,b\}.\) Determine whether \( \land \) on \(\mathbb{Z}\),
- is closed,
- is commutative,
- is associative, and
- has an identity.
Exercise \(\PageIndex{7}\): Min
For \(a, b \in \mathbb{Z},\) define an operation \( \lor \), by \( a\lor b= min \{a,b\}.\) Determine whether \( \lor \) on \(\mathbb{Z}\),
- is closed,
- is commutative,
- is associative, and
- has an identity.
Exercise \(\PageIndex{8}\): Defection
For \(a, b \in \mathbb{Z},\) define an operation \( \star_5 \), by \( a\star_5 b= a+b-5 .\) Determine whether \( \star_5 \) on \(\mathbb{Z}\),
- is closed,
- is commutative,
- is associative, and
- has an identity.
- Answer
-
is closed, is commutative, is associative, and has an identity.
Exercise \(\PageIndex{9}\): Distributive
Determine whether \( \otimes \) is distributive over \( \oplus \) on \( \mathbb Z\).
- Answer
-
Counterexample:
Choose \(a = 2, b=3, c=4.\)
Then consider \( 2 \otimes (3 \oplus 4)=2 \otimes [(3)(4) + 3 + 4] =2 \otimes 19=(2 + 19)(2 + 19) =441. \)
Now consider \((2 \otimes 3) \oplus (2 \otimes 4)=[(2 + 3)(2 + 3)] \oplus [(2 + 4)(2 + 4)]= 25 \oplus 36 =(25)(36)+25+36 =961.\)
Since \(441 \ne 961,\) the binary operation \( \otimes \) is not distributive \( \oplus \) overon \( \mathbb Z\). ⬜
Exercise \(\PageIndex{10}\): Distributive
Determine whether \( \oslash\) is distributive over \( \oplus \) on \( \mathbb Z\).
Exercise \(\PageIndex{11}\): Even Multiplication
Let \(S=\{1, 2,4,6, \dots \} \), that is \(S\) is the set of all even positive integers and \(1\). Use the multiplication \( \times \) as the binary operation.
Determine whether the multiplication \( \times \) on \(S\),
- is closed,
- is commutative,
- is associative, and
- has an identity.
Exercise \(\PageIndex{12}\): Arithmetic operations on Rationals and Irrationals
- Is addition closed on \( \mathbb Q\)?
- Is multiplication closed on \( \mathbb Q\)?
- Is addition closed on \( \mathbb Q^c\)?
- Is multiplication closed on \( \mathbb Q^c\)?