NPTest1solutions - Numbers and Polynomials Test 1 solutions...

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Test 1 solutions Answer FOUR questions. Be sure to give a reason for each step; if that reason is one of the axioms, identify which one by its symbol. 1. Show that if a R then 0 a = 0. 2. Let a, b R . (i) Show that ( - a ) b = - ( ab ) = a ( - b ). (ii) Show that ( - a )( - b ) = ab . 3. Let a, b R . (i) Show that a 2 + b 2 > 0. (ii) Show that if a 2 + b 2 = 0 then a = b = 0. 4. Show that if a R and if b R is nonzero then ± ± ± a b ± ± ± = | a | | b | . Note : It may be assumed that x > y > 0 x 2 > y 2 . 5. Define N as a subset of R and prove that no element n of N satisfies 0 < n < 1. Solutions : 1. Note first that 0 a AID = (0 + 0) a D = 0 a + 0 a whence addition of - 0 a yields 0 AIV = ( - 0 a )+0 a = ( - 0 a )+(0 a +0 a ) AA = (( - 0 a )+0 a )+0 a AIV = 0+0 a AID = 0 a and concludes the proof. 2. (i) Note that ab + ( - a ) b D = ( a + ( - a )) b AID = 0 b 1 = 0 so that ( - a ) b is indeed the additive inverse - ( ab ) of ab ; similarly, a ( - b ) equals - ( ab ). 1
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NPTest1solutions - Numbers and Polynomials Test 1 solutions...

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