Assignment 1 : Problem 4a

[Aaron Albers]

For a,b  N, define a ~ b iff a^2 + b^2 is even

Set A = { (a, b)  R | a^2 + b^2 is even }

Reflexive: { (a, a)  R | a^2 + a^2 is even }

Symmetric: { (b, a)  R | b^2 + a^2 is even } 

Transitive: { (a, c)  R | a^2 + c^2 is even }  &&  Symmetric

Something to start with....

[Alex]

1.     For a, b ∈ N, define a ∼ b if and only if a² + b² is even.

a.      Prove that ∼ defines an equivalence relation on N.

                                               i.     Reflexive: if { a ∈ N | a² is even }, then a² = a² so a ~ a. Then the relation is reflexive.

                                              ii.     Symmetric: If a ~ b a² = b² so b² = a² and b ~ a.  The relation is symmetric.

                                            iii.     If a ~ b and b ~ c then a² = b² and b² = z².  Thus a = b = c.  since a = c, a ~ c, so the relation is transitive.

[anne speck]

The formula in the book is a² + b. She said a² + b²
was a typo and the book takes precedence.

[Aaron Albers]

I agree... seeing also that (a^2 + b^2 is even) is not reflexive therefore not an equivalence.