Worksheets — Set 9

Abstract algebra · Rings and fields

Print and write your answers. Use diagrams to show thinking.

Warm-up

1. Prove: the sum of two even integers is even. ___ 2. Give a direct proof that √12 is irrational (outline key step). ___ 3. Prove by induction: 1 + 2 + … + n = n(n+1)/2

_See printable PDF for diagram._

Written practice

1. Prove: the sum of two even integers is even. ___ 2. Give a direct proof that √11 is irrational (outline key step). ___ 3. Prove by induction: 1 + 2 + … + n = n(n+1)/2 4. State the contrapositive of: if n² is even, then n is even ___ 5. Find a counterexample if false: every continuous f is differentiable 6. Write a ε-δ style argument outline for a limit. ___ 7. Give a direct proof that √14 is irrational (outline key step). ___ 8. Give a direct proof that √17 is irrational (outline key step). ___

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More written practice

1. Give a direct proof that √20 is irrational (outline key step). ___ 2. Give a direct proof that √23 is irrational (outline key step). ___ 3. 6 + 7 = ___ 4. 2 + 4 = ___ 5. 3 + 5 = ___ 6. 4 + 6 = ___

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Patterns and rules

1. Prove: the sum of two even integers is even. ___ 2. Give a direct proof that √28 is irrational (outline key step). ___ 3. Prove by induction: 1 + 2 + … + n = n(n+1)/2

_See printable PDF for diagram._

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Parent tip: Start from what is given