## Transcribed Text

1. Use only the field and order axioms to prove the following statements for .2, y, Z € R.
Make sure you indicate which axiom you are utilizing at each step.
(a) Prove that if r > 0 and y > 0, then x + > 0.
(b) Prove that if r > 0 and y > 0, then xy > 0.
(c) Prove that for z + 0 and y + O,then
(d) Prove that 0 is its own additive inverse, i.e. 0 - -0.
(c) Prove that 1 is its own multiplicative inverse, i.e. 1
(f) Prove that r - y - 0 iff r - y.
(g) Prove that if xy - Z and any two of x,y and Z are positive, then the third real
number must also be positive.
(h) Prove that if . > 0 then also =-1 > 0.
2. Given that
[ - { -r, x, if 0 if x 0
prove the following statements for z, y € IR :
(a) 0 for all I € IR: and - 0 iff 7 - 0.
(b) [ - 1-z%
(c) - I <
(d) S y - -y S Z y.
(e) x - y .
(f) ||x| - [y| 1s/2+y/.
3. For the following sets A, find (if they exist) max A, min A, (sup A, inf (A) :
(a) A - {1,3,9,4,0} .
(b) A - [0,00).
(c) A - [-1,3] -
(d) A - {x:-2 - 1 - -0}.
(e) A - {x R: I2 < 2}.
(f) A - (-00,00) .
(g) A - } -

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