{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

NPTest1solutions

# NPTest1solutions - Numbers and Polynomials Test 1 solutions...

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 2

NPTest1solutions - Numbers and Polynomials Test 1 solutions...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online