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Basic Proof Writing

  1. May 29, 2015 #1
    1. The problem statement, all variables and given/known data
    Given a series of mathematical statements, some of which are true and some of which are false. Prove those of which are true, and disprove those which are false.
    1. The sum of three odd integers is odd.
    Text: Principles of Mathematics by Allendoefer and Oakley.



    2. Relevant equations
    x(px→qx)↔P⊆Q
    Law of Detachment and Law of Substitution.


    3. The attempt at a solution
    I am mainly looking for feedback on my notation (formalism). What is the proper way of writing out a proper basic proof?
    Statement: The sum of three odd integers is odd.
    x(integers): If x is odd, then ∑3i=1xi is odd.
    (1) Assume x1, x2, and x3 is odd. [Hypothesis]
    (2) The integers a, b, and c exist such that x1=2a+1, x2=2b+1, and x3=2c+1. [Defn of odd]
    (3) ∑3i=1xi =(2a+1)+(2b+1)+(2c+1)
    =2a+1+2b+1+2c+1
    =2a+2b+2c+2+1
    =2(a+b+c+1)+1
    =2(m)+1[Let m=a+b+c+1]
    Therefore the sum of three odd integers is odd by definition.
     
  2. jcsd
  3. May 29, 2015 #2

    Mark44

    Staff: Mentor

    Not wrong, but writing ##\sum_{i = 1}^3 x_i## seems like overkill here. Just let x, y, and z be the odd integers. Their sum is x + y + z.

    The gist of your proof is fine, but you're using something that is in my opinion unnecessary (the summation and subscripted variables).
     
  4. May 29, 2015 #3
    Thank for replying Mark44 and taking the time to help me. Would the following adjustment make the proof less overkill?

    Statement: The sum of three odd integers is odd.
    xyz(integers): If x, y, and z are odd, then x+y+z is odd.
    (1) Assume x, y, and z are odd [Hypothesis]
    (2) The integers a, b, and c exist such that x=2a+1, y=2b+1, and z=2c+1. [Defn of odd]
    (3) x+y+z =(2a+1)+(2b+1)+(2c+1)
    =2a+1+2b+1+2c+1
    = 2a+2b+2c+2+1
    =2(a+b+c+1)+1
    =2(m)+1 [Defn of odd]
    Therefore x+y+z is odd by definition of odd.
     
  5. May 30, 2015 #4

    Mark44

    Staff: Mentor

    That looks good. The only think I would add is that it wouldn't hurt (IMO) to reduce the formalism a bit more in your statement of the problem. Instead of saying "∀x∀y∀z(integers): If x, y, and z are odd, then x+y+z is odd," you could say the same thing as "For any integers x, y, and z, x + y + z is odd." All the rest looks fine.
     
  6. May 30, 2015 #5
    Thanks for the feed back Mark44. Going forward I will have more questions about this topic. Should I keep creating new threads, as in a new thread per question or just keep everything under one thread? Thanks.
     
  7. May 30, 2015 #6

    Mark44

    Staff: Mentor

    New question -- new thread.
     
  8. May 30, 2015 #7
    Gotcha, see you there haha.
     
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