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Spivak Ch1, Q8: Deducing Basic Properties of Numbers

  1. Jul 29, 2017 #1
    1. The problem statement, all variables and given/known data
    Although the basic properties of inequalities were stated in terms of the collection
    P of all positive numbers, and < was defined in terms of P, this
    procedure can be reversed. Suppose that P10-P12 are replaced by
    (P'10) For any numbers a and b one, and only one, of the
    following holds:
    (i) a = b,
    (ii) a < b,
    (iii) b < a.
    (P'11) For any numbers a, b, and c, if a < b and b < c, then
    a < c.
    (P'12) For any numbers a, b, and c, if a < b, then
    a+c<b+c.
    (P'13) For any numbers a, b, and c, if a < b and 0 < c, then
    ac < bc.
    Show that P10-P12 can then be deduced as theorems.

    2. Relevant equations
    Addition properties:

    P1: (Associative law of addition) (a+b)+c=a+(b+c)
    P2: (Existence of additive identity) a+0=0+a=a
    P3: (Existence of additive inverse) a+(-a)=(-a)+a=0
    P4: (Commutative law of addition) a+b=b+a

    Multiplication properties:
    P5: (Associative law of multiplication) a x (b x c)=(a x b) x c
    P6: (Existence of multiplicative identity) a x 1=1 x a=a
    P7: (Existence of multipliative inverse) a x a-1=a-1 x a=1 (If a is different from zero)
    P8: (Commutative law of multiplication) a x b=b x a
    Distributive property
    P9: (Distributive law) a x (b+c)=(a x b)+(a x c)

    Order properties:
    Let P be called the collection of positive numbers, then:
    P10: (Trichotomy Law) For every a, exactly one of the following happens: a=0, a is in P or -a is in P
    P11: (Closure under addition) If a and b are in P, a+b is in P
    P12: (Closure under multiplication) If a and b are in P, a x b is in P

    3. The attempt at a solution
    I'm really not entirely sure what the question is asking me to do. I believe it is asking me to show that P'10-P'13 are true from the properties P1-P9 and thus the statements in P10-P12 are also true.

    Here is my attempt at doing that.

    Starting with:

    (P'12) For any numbers a, b, and c, if a < b, then
    a+c<b+c.
    From P3, if a<b, then a+(-b)<b+(-b)
    -> a-b<0
    Thus a-b+c-c<0
    ->(a+c)+(-b-c)<0
    ->(a+c)-(b+c)+(b+c)<0+(b+c)
    ->a+c<b+c
    Thus we have deduced P'12

    Now P'11:
    (P'11) For any numbers a, b, and c, if a < b and b < c, then
    a < c.

    From P'12, a+c-b-c<0
    ->a+c-b<c
    c-b>0, then (c-b)+a>0+a
    thus a<a+c-b<c
    Therefore we have deduced P'11.

    I'm fairly confident in these 2, the next 2 are where I don't feel as though my proof is very good.

    P'10:
    (P'10) For any numbers a and b one, and only one, of the
    following holds:
    (i) a = b,
    (ii) a < b,
    (iii) b < a.

    Using P7, if a<b then 1<b/a, which would mean 1<1 if a=b. From (P3) a+(-a)=0, thus b+a+(-a)=0+b. So 1<1 and 1=1 cannot both be true and thus if a<b then it can't also be true that a=b.

    Similarly for b<a, then 1<a/b, implies 1<1 if a=b. So i cannot be true if either ii or iii are true. Lastly if b<a is true and a<b is true then both b-a<0 and a-b<0 would have to be true.

    Assuming a>0 and b>0, then from P9, a(b-a)<0 and a(a-b)<0 are both true, thus -a^2+ab<0->a^2>ab and from a(a-b)<0 a^2<ab. a^2<ab<a^2 cannot be true unless a=0.

    Similarly we can find that b^2<ab<b^2 would only be true if b=0. Thus if both ii and iii were true it would cause contradictions in 0=0 and 0<0 and a=b, which we've already established cannot be true. So i, ii and iii are exclusive and P'10 is true.

    P'13.
    (P'13) For any numbers a, b, and c, if a < b and 0 < c, then
    ac < bc.

    a<b->a-b<0<c. Using P9, c(a-b)<0<c(c)
    -> ca-cb<0<c^2.
    Then applying P3, ac-bc+bc<0+bc
    -> ac<bc
    Therefore we have deduced P'13.

    For P10, since we've shown that either a=b, a<b or b<a. Then if a=0 clearly b<a or (-b)<a as b=/=a and one of either b<a or (-b)<a must be true. So any number x must either be a=0, a<0 or (-a)>0.

    I believe I still must somehow show that P11 and P12 statements are true somehow and then from that I would simply state that the set P can represent all numbers a>0 or (-a)>o, if a<0.

    That's basically what I got, again I'm not entirely sure that I've understood the question properly.
     
  2. jcsd
  3. Jul 30, 2017 #2

    vela

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    You're being asked to assume P1-P9 and P'10-P'13 are true and prove that P10-P12 follow as a consequence.


    You can't say that a<b implies a+(-b) < b+(-b) because that implication is exactly what you're setting out to prove here.

    But as I said, that's not what the problem is asking you to do anyway.
     
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