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Mathematics LibreTexts

2 E: Exercises

  • Page ID
    7429
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    Exercise \(\PageIndex{1}\):

    Determine whether or not each of the following binary relations \(R\) on the given set \(A\) is reflexive, symmetric,
    antisymmetric, or transitive. If a relation has a certain property, prove this is so; otherwise, provide a counterexample
    to show that it does not.  If \(R\) is an equivalence relation, describe the equivalence classes of \(A\).

    1.  Let \(S=\{0,1,2,3,4,5,6,7,8,9 \}.\) Define a relation \(R\) on \(A= S \times S\) by  \((a,b)\, R\, (c,d)\) if and only if \( 10 a +b \leq 10 c+ d.\)

    2.  Let \(A= \mathbb{Z} \backslash \{0\}.\) Define a relation \(R \) on \(A,\)  by  \(a \,R\, b\)  if and only if \( ab>0.\)

    3. Define  a relation \(R\) on \(A={\mathbb Z}\) by \(a \,R\, b\) if and only if  \( 4 \mid (3a+b).\)
    4.  Define  a relation \(R\) on \(A={\mathbb Z}\) by \(a \, R \, b\) if and only if  \( 3 \mid (a^2-b^2).\)

    5. Let \(A=\mathbb{R}\), If \(a.b \in \mathbb{R}\), define  \(a \, R \, b\) if and only if  \( a-b \in \mathbb{Z}.\)

    6. Define a relation \(R\) on the set \({\mathbb Z} \times{\mathbb Z}\)  by: \((a,b)\, R \,(c,d) \mbox{ if and only if } ac=bd.\)

    7. Define a relation \(R\) on \({\mathbb Z}\) by: \(a R b\) if and only if \( 5 \mid 2a+3b.\)

    8.   Define a relation \(R\) on \({\mathbb Z}\) by: \(a R b\) if and only if \( 2 \mid a^2+b.\)

    Exercise \(\PageIndex{2}\):

    Let \(a, b,c, d \in \bf Z_+.\)

    1.   If \(a|b\) and \(a|c\) , is it necessarily true that \(a|(b + c)?\)
    2.   If \(a|(b + c)\), is it necessarily true that \(a|b\) and \(a|c\)?
    3.  If \(a|bc\),  is it necessarily true that \(a|b\) and \(a|c\)?
    4.   If \((a+b)|c\),  is it necessarily true that \(a|c\) and \(b|c\)?
    5. If \(a|c\) and \(b|c\) , is it necessarily true that \((a+b)|c?\)
    6.   If \(a^3|b^4\), is it necessarily true that \(a|b.\)
    7. If \(a|b\) , is it necessarily true that \(a^3 \mid b^5?\)
    8.   If \(c|a\) and \(d|b\), is it necessarily true that \(cd|ab\).

     

    Exercise \(\PageIndex{3}\):

    1. Find  all possible values for the missing digit if \(12345X51234\) is divisible by \(3.\)
    2.  Using divisibility tests, check if the number \(355581\) is divisible by \(7\)
    3.   Using divisibility tests, check if the number \(824112284\) is divisible by \(5, 4,\) and \(8.\)

     

    Exercise \(\PageIndex{4}\):

    Let \(a\) and \(b\) be positive integers such that \(7|(a+2b-2)\) and \( 7|(b-9).\) Prove that \(7|(a+b).\)

     

    Exercise \(\PageIndex{5}\):

    1. In a \(113\)-digit multiple of \(13\), the first \(56\) digits are all \(5\)s and the last \(56\) digits are all \(8\)s. What is the middle digit?
    2. In a \(113\)-digit multiple of \(7\), the first \(56\) digits are all \(8\)s and the last \(56\) digits are all \(1\)s. What is the middle digit?

     

    Exercise \(\PageIndex{6}\):

    Prove the statements that are true and give counterexamples to disprove those that are false.
     Let \(a,b\) and \(c\) be integers.

    1. If \(a|b\) then \(b|a\).
    2.  If \(a|bc\) then \(a|b\) and \(a|c.\)
    3.  If \(a|b\) and \(a|c\) then \(a|bc\).
    4. If \(a|b\) and \(a|c\) then \(a|(b+c)\) and \(a|(b-c)\).
    5.  If \(a|(b+c)\) and \(a|(b-c)\) then \(a|b\) and \(a|c\).
    6.  If \(a|b\) and \(a|c\) then \(a|(b+c)\) and \(a|(2b+c)\).
    7. If \(a|(b+c)\) and \(a|(2b+c)\) then \(a|b\) and \(a|c\)

     

    Exercise \(\PageIndex{7}\)

    Prove that for all integers \( n\geq 1, \,5^{2n}-2^{5n}\) is divisible by \(7.\)

    Answer

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