Algebra/Chapter 2/Logic and Proofs

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Algebra
Algebra/Chapter 2
 ← Real Numbers Logic and Proofs

2.6: Logic and Proofs


Even and Odd Numbers[edit | edit source]

Contructing a Proof of a Conditional Statement[edit | edit source]

Properties of Equality[edit | edit source]

Property Name Addition Subtraction Multiplication Division
Commutative Doesn't work:

This does:
Doesn't work:

This does:
Associative Doesn't work:

This does:
Doesn't work:

This does:
Identity
Inverse   as long as a ≠ 0.   as long as a ≠ 0.
Distributive
But wait:

Practice Problems[edit | edit source]

Problem 2.80 (Using Properties of Numbers) Justify each step, using the properties of communativity and associativity in proving the following identities.








Problem 2.81 (Using Properties of Numbers) Determine if the following statements are true or false. Justify your conclusions.

a. If , , and are integers, then the number is an even number.
b. If and are odd integers, and is an integer, then the number is an even number.

Problem 2.82 (Using Properties of Numbers) We define an integer to be of

  • Type I if for some integer
  • Type II if for some integer
  • Type III if for some integer
  • Type IV if for some integer

a. Provide at least two examples of each of the four types of integers above.
b. Is it true that if is even, then it is of type I or III? Justify your answer.
c. Is it true that if is of type I, whenever or are of type III? Justify your answer.

Problem 2.83 (Using Properties of Numbers) For all real numbers and positive integers , show that: