Proving a partial order and total order.

In each of the following say whether or not R is a partial order on A. If so, is it a total order? a) A= {a,b,c,d}, R= {(a,a),(b,a),(b,b),(b,c),(c,c)} b) A is the set of positive divisors of 24, that is A= {1,2,3,4,6,8,12,24}, and the relation R is dividing. If B is the set of positive divisors of 24 except 1, what is the minimal element(s)? smallest element? maximal element(s)? largest element of B?

Proving a partial order and total order.

Say whether or not R is a partial order and a total order on A. Show proof. A= {a,b,c}, R= {(a,a),(b,a),(b,b),(b,c),(c,c)

Discrete Math

Discrete math questions. Please provide formulas and all calculations for all 22. They are very short answer type questions.

Equivalence Relation, equivalence classes and the partition defined by the relation.

Verify the relation R on Z defined by xRy iff 5(x-y), i.e., 5 divides x-y, is an equivalence relation. Describe the equivalence classes, and the partition defined by this equivalence relation.

Discrete Math: Mathematical Induction

Please see the attached file for the fully formatted problem. Without using Theorem 4.2.2, use mathematical induction to prove that P(n): 1 + 5 + 9 + ... + (4n - 3) = n(2n - 1) for all integers n >= 1