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Thread: Domain and Range Notation with equation: r = {(x, y) ∈ I x I : y = x - 2}

  1. #1

    Domain and Range Notation with equation: r = {(x, y) ∈ I x I : y = x - 2}

    When you are writing domain and rage in set notation, can you write the range as the equation? Here's what I mean specifically:


    Find the domain and range for r={(x,y)∈I x I : y=x-2}. Write in set builder notation.


    Domain={x : x∈I}
    Range = {y : y=x-2}

    Can you write the range like that?
    Last edited by mmm4444bot; 11-15-2017 at 03:11 AM. Reason: Auto-smiley issue

  2. #2
    Elite Member mmm4444bot's Avatar
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    Quote Originally Posted by thunc14 View Post
    Find the domain and range for r={(x,y)∈I x I : y=x-2}. Write in set builder notation.

    Domain={x : x∈I}
    Range = {y : y=x-2}

    Can you write the range like that?
    I guess it's true, but set-builder notation doesn't normally mention relationships between different variables; instead, it lists set(s) or intervals of possible values for the variable that appears to the left of the colon.

    You list Integers, as the set of all possible x-values. What is the set of all possible y-values?
    "English is the most ambiguous language in the world." ~ Yours Truly, 1969

  3. #3
    Quote Originally Posted by mmm4444bot View Post
    I guess it's true, but set-builder notation doesn't normally mention relationships between different variables; instead, it lists set(s) or intervals of possible values for the variable that appears to the left of the colon.

    You list Integers, as the set of all possible x-values. What is the set of all possible y-values?

    Thank you for that explanation, makes a lot of sense. Much appreciated!

  4. #4
    Quote Originally Posted by mmm4444bot View Post
    I guess it's true, but set-builder notation doesn't normally mention relationships between different variables; instead, it lists set(s) or intervals of possible values for the variable that appears to the left of the colon.

    You list Integers, as the set of all possible x-values. What is the set of all possible y-values?
    Also, for the domain notation, can you just write {x∈I} instead of {x : x∈I}?
    Last edited by mmm4444bot; 11-15-2017 at 04:12 AM. Reason: Auto-smiley issue

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    Elite Member mmm4444bot's Avatar
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    Quote Originally Posted by thunc14 View Post
    Also, for the domain notation, can you just write {x∈I} instead of {x : x∈I}?
    If the instructions ask for Set-Builder Notation, the answer is no (for both the domain and the range). Otherwise, you could report:

    Domain: x ∈ ℤ

    Range: y ∈ ℤ

    ℤ is also a symbol for representing the set of Integers.
    "English is the most ambiguous language in the world." ~ Yours Truly, 1969

  6. #6
    Quote Originally Posted by mmm4444bot View Post
    If the instructions ask for Set-Builder Notation, the answer is no (for both the domain and the range). Otherwise, you could report:

    Domain: x ∈ ℤ

    Range: y ∈ ℤ

    ℤ is also a symbol for representing the set of Integers.

    I apologize, I find that answer a little confusing. You say no I can't write domain = {x ∈ I} instead of the correct way domain = {x: x ∈ I}, but what I can do is write domain: x ∈ Z, where Z is the set of integers?

    I'm not sure what I am misunderstanding.

    Thank you so much for your help.

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    Quote Originally Posted by thunc14 View Post
    I apologize, I find that answer a little confusing. You say no I can't write domain = {x ∈ I} instead of the correct way domain = {x: x ∈ I}, but what I can do is write domain: x ∈ Z, where Z is the set of integers?

    I'm not sure what I am misunderstanding.

    Thank you so much for your help.
    It isn't clear what you think you misunderstand ... as far as I can see, you are correctly interpreting what was said.

    "Set-builder notation" is very specific; it always has the form "{x: some condition on x}", so if the instructions for the problem specified this form, then you can't write "{x ∈ I}".

    If the problem just said "set notation", that covers several different forms, so you would have more freedom; but "{x ∈ I}" still is not a valid form. On the other hand, as far as I am concerned, "x ∈ I" (or "x ∈ Z") is not exactly a valid answer either: it is a statement about x, not a definition of a set. To identify the domain itself, you could just say "domain = I"!

    Perhaps your confusion is about the difference between "{x ∈ I}" and "x ∈ I". The former would mean "the set containing the element 'x ∈ I' ", which is sort of silly; the latter is the statement, "x is an element of the set of integers".

    But since, as you stated the problem, it did call for set-builder notation, the proper answer is "domain = {x: x ∈ I}". Most of what I've said here is just confusing quibbles, and can be safely ignored.

  8. #8
    Quote Originally Posted by Dr.Peterson View Post
    It isn't clear what you think you misunderstand ... as far as I can see, you are correctly interpreting what was said.

    "Set-builder notation" is very specific; it always has the form "{x: some condition on x}", so if the instructions for the problem specified this form, then you can't write "{x ∈ I}".

    If the problem just said "set notation", that covers several different forms, so you would have more freedom; but "{x ∈ I}" still is not a valid form. On the other hand, as far as I am concerned, "x ∈ I" (or "x ∈ Z") is not exactly a valid answer either: it is a statement about x, not a definition of a set. To identify the domain itself, you could just say "domain = I"!

    Perhaps your confusion is about the difference between "{x ∈ I}" and "x ∈ I". The former would mean "the set containing the element 'x ∈ I' ", which is sort of silly; the latter is the statement, "x is an element of the set of integers".

    But since, as you stated the problem, it did call for set-builder notation, the proper answer is "domain = {x: x ∈ I}". Most of what I've said here is just confusing quibbles, and can be safely ignored.
    Forgive my ignorance here, but my confusion can be summed up by asking for a breakdown in the difference of these 4 notations regarding the equation:

    Domain = {(x: x ∈ I)} vs. Domain = {x ∈ I} vs. Domain = Z vs. Domain = {Z} vs. Domain = (-∞,∞)?
    Last edited by thunc14; 11-16-2017 at 03:09 AM.

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    Elite Member mmm4444bot's Avatar
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    Quote Originally Posted by thunc14 View Post
    my confusion can be summed up by asking for a breakdown in the difference of these 4 notations regarding [domain statements]:
    Posting a question is not a summation. (By the way, you have asked about five forms, not four.)

    I'm not sure what kind of "breakdown" you want. Except for the last one, they are all different forms of saying the same thing (after corrections, noted below).


    {(x: x ∈ I)}
    If you remove the unneccessary parentheses, what remains is called Set-Builder Notation.

    Here, your symbol I represents "the set of Integers".

    We read this notation as, "The set of all x such that x is an Integer".


    {x ∈ I}
    If you remove the unneccessary curly braces, what remains is a simple statement: "x is an Integer".


    Z
    Here, symbol represents "the set of Integers".


    {Z}
    This form is the same as the previous one. The curly braces are unneccessary. Symbol already represents a set.

    Putting curly braces around it leads to: "the set of the set of Integers". That's redundant.


    (-∞,∞)
    This form is called Interval Notation. It cannot be used to express the set of Integers.

    What you have typed is read as, "all Real numbers".

    The symbol for that set is .


    PS: In addition to symbol for the set of Integers and symbol for the set of Real numbers, the following double-stroke characters represent other common sets:

    The set of Natural numbers:

    The set of Rational numbers

    The set of Complex numbers:

    Last edited by mmm4444bot; 11-17-2017 at 03:29 AM. Reason: Agree with DrPeterson; added "after corrections, "
    "English is the most ambiguous language in the world." ~ Yours Truly, 1969

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    Quote Originally Posted by thunc14 View Post
    Forgive my ignorance here, but my confusion can be summed up by asking for a breakdown in the difference of these 4 notations regarding the equation:

    Domain = {(x: x ∈ I)} vs. Domain = {x ∈ I} vs. Domain = Z vs. Domain = {Z} vs. Domain = (-∞,∞)?
    Note that mmm4444bot, in saying these mean the same thing, first had to correct them, because in fact three of them do NOT mean what you think they do.


    • The parentheses he removed from the first are not permitted -- they are not just optional!
    • I already discussed {x ∈ I} as improper notation.
    • And {Z} would actually mean "a set consisting of the set of integers", which is NOT the same as "the set of integers", just as "a bag containing a bag of lemons" is not the same as "a bag containing lemons". The only element of {Z} is the set Z.


    Set notation is very precise, and can't be modified without care.

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