Question

1.

a. Consider the definition of relation. If A is the set of even numbers and ≡ is the subset of ordered pairs (a,b) where a<b in the usual sense, is ≡ a relation? Explain.

b. Consider the definition of partition on the bottom of page 18. Theorem 2 says that the equivalence classes of an equivalence relation form a partition of the set. Consider the set ℕ with the equivalence relation ≡ defined by the rule: a≡b in ℕ if a,b have the same remainder when divided by 5. For example, a=1 and b=6 works since both 1 and 6 have the same remainder when divided by 5. Write down the equivalences classes for this equivalence relation. Verify that they do partition the set ℕ (HINT: to form one of the classes, think about all the numbers that are equivalent to 1 and 6)

Answer 1