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Math Help - How's that div and mod are functions on Cartesian products of integer?

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    Senior Member x3bnm's Avatar
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    How's that div and mod are functions on Cartesian products of integer?

    We know that cartesian product of two sets A = \{a,b,c\} and B = \{d,e,f\} is:

     A \times B\ =\ \{(a,d),(a,e),(a,f),(b,d),(b,e),(b,f),(c,d),(c,e),  (c,f)\}

    and
     m\ \mathbf{div}\ n\ = the integer quotient obtained when m is divided by n

     a\ \mathbf{mod}\ b\ = the integer remainder obtained when a is divided by b


    I'm reading a Discrete math book and I am on Functions chapter. It has a line in it that says:

    " \mathbf{mod} and \mathbf{div} are really functions defined on Cartesian products of integers."


    For what reason is this statement true? I know it's true but I've no idea why.

    I don't see the connection between Cartesian product and the \mathbf{mod}, \mathbf{div} functions.

    Can anyone kindly tell me when I use \mathbf{mod} and \mathbf{div} functions how is that they are really functions on Cartesian products of integers?
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    Re: How's that div and mod are functions on Cartesian products of integer?

    Quote Originally Posted by x3bnm View Post
    We know that cartesian product of two sets A = \{a,b,c\} and B = \{d,e,f\} is:
     A \times B\ =\ \{(a,d),(a,e),(a,f),(b,d),(b,e),(b,f),(c,d),(c,e),  (c,f)\}
    and
     m\ \mathbf{div}\ n\ = the integer quotient obtained when m is divided by n
     a\ \mathbf{mod}\ b\ = the integer remainder obtained when a is divided by b
    I'm reading a Discrete math book and I am on Functions chapter. It has a line in it that says: " \mathbf{mod} and \mathbf{div} are really functions defined on Cartesian products of integers."
    Can anyone kindly tell me when I use \mathbf{mod} and \mathbf{div} functions how is that they are really functions on Cartesian products of integers?
    When posting this sort of question, it is useful to say what book.
    I guess that from the way that textbook is using those words, both \mod~\&~div assign pairs of integers to an integer.
    So in that sense we have \mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}.
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    Senior Member x3bnm's Avatar
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    Re: How's that div and mod are functions on Cartesian products of integer?

    Quote Originally Posted by Plato View Post
    When posting this sort of question, it is useful to say what book.
    I guess that from the way that textbook is using those words, both \mod~\&~div assign pairs of integers to an integer.
    So in that sense we have \mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}.
    Thank you for your answer. I didn't give the name of book. Sorry about that.

    I'm reading Susanna S. Epp's Discrete Mathematics with Applications(Second Edition).


    So if 3 = \{1,2,3\} and 2 = \{1^\prime, 2^\prime\}

    3\ \times\ 2 = \{(1, 1^\prime),(1, 2^\prime), (2, 1^\prime),(2, 2^\prime),(3, 1^\prime), (3, 2^\prime)\}

    For \mathbf{mod} and \mathbf{div} function what do you assign here?

    There are 6 elements of 3 \times 2. How do you calculate div and mod here?

    Can you kindly explain on asssigning an integer to \mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}? Sorry for not understanding.
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    Senior Member x3bnm's Avatar
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    Re: How's that div and mod are functions on Cartesian products of integer?

    Sorry for reopening this thread. I thought I understood the problem. Sorry for jumping to conclusion so quickly.

    @Plato:

    Can you kindly elaborate on what you said about "assign pairs of integers to an integer" comment?
    Last edited by x3bnm; August 1st 2011 at 12:56 PM.
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    Re: How's that div and mod are functions on Cartesian products of integer?

    Quote Originally Posted by x3bnm View Post
    So if 3 = \{1,2,3\} and 2 = \{1^\prime, 2^\prime\}

    3\ \times\ 2 = \{(1, 1^\prime),(1, 2^\prime), (2, 1^\prime),(2, 2^\prime),(3, 1^\prime), (3, 2^\prime)\}

    For \mathbf{mod} and \mathbf{div} function what do you assign here?
    That doesn't make sense.

    The book says that mod and div are functions defined on this cartesian product : \mathbb Z \times \mathbb Z (or a subset).
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    Re: How's that div and mod are functions on Cartesian products of integer?

    Quote Originally Posted by x3bnm View Post
    Sorry for reopening this thread. I thought I understood the problem. Sorry for jumping to conclusion so quickly.
    @Plato:
    Can you kindly elaborate on what you said about "assign pairs of integers to an integer" comment?
    I no longer have a copy of Epp's book. As I recall, wrote a positive review to the publisher. Because I have a positive impression of that text some of wording you posted worries me.
    For the mod operator, it is clear that one could apply that operator to any pair integers and the output is an integer.
    Examples from MathCad: \mod(17,3)=2,~\mod(14,-3)=2,~\mod(3,6)=3.

    But on the other hand, there is no similar understanding for what Div could mean. Clearly it is not true that any integer divides every integer. So how exactly does she define the Div operator?
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    Senior Member x3bnm's Avatar
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    Re: How's that div and mod are functions on Cartesian products of integer?

    Quote Originally Posted by Plato View Post
    I no longer have a copy of Epp's book. As I recall, wrote a positive review to the publisher. Because I have a positive impression of that text some of wording you posted worries me.
    For the mod operator, it is clear that one could apply that operator to any pair integers and the output is an integer.
    Examples from MathCad: \mod(17,3)=2,~\mod(14,-3)=2,~\mod(3,6)=3.

    But on the other hand, there is no similar understanding for what Div could mean. Clearly it is not true that any integer divides every integer. So how exactly does she define the Div operator?
    Now it makes sense. Thank you for your response. I agree with you about the statement. Not all integers divide other integers.

    So the statement is rather vague. No offense.

    But I like this book. The author explained things here like teaching an elementary school kid.
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    Re: How's that div and mod are functions on Cartesian products of integer?

    Quote Originally Posted by x3bnm View Post
    But I like this book. The author explained things here like teaching an elementary school kid.
    If that is important to you may I suggest Discrete Mathematics for Teachers
    I did an Eisenhower Institute as a test site for that text.
    IMO it was a huge success with the in service high school teachers.
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    Senior Member x3bnm's Avatar
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    Re: How's that div and mod are functions on Cartesian products of integer?

    Quote Originally Posted by Plato View Post
    If that is important to you may I suggest Discrete Mathematics for Teachers
    I did an Eisenhower Institute as a test site for that text.
    IMO it was a huge success with the in service high school teachers.
    I'll look into the links. Again thanks.
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