6. Prove the following relation:
Prove that the following relation R is an equivalence relation on the set of ordered pairs of real numbers. Describe the equivalence classes of R. (x, y)R(w, z) y-x2 = z-w2
9. Let R an equivalence relation. Prove or disprove that R:R is an equivalence relation
Prove the following relation for ARMA(2,1) 2 P1P2 P3 2 pi-P2
1. Define a relation on Z by aRb provided a -b a. Prove that this relation is an equivalence relation. b. Describe the equivalence classes. 2. Define a relation on Z by akb provided ab is even. Use counterexamples to show that the reflexive and transitive properties are not satisfied 3. Explain why the relation R on the set S-23,4 defined by R - 11.1),(22),3,3),4.4),2,3),(32),(2.4),(4,2)) is not an equivalence relation.
Prove that if R is an equivalence relation on a set A, then R ^-1 is an equivalence relation on A.
2. (5pt) Consider the following binary relations. In each case prove the relation in question is an equivalence relation and describe, in geometric terms, what the equivalence classes are. (a) Si is a binary relation on R2 x R2 defined by z+ly-+ 1 r,y). (,y) e S Recall that R =R x R. (b) Sa is a binary relation on R defined by 1-ye2 r,y) e S
Prove or disprove: The relation "is-a-normal-subgroup-of" is a transitive relation. (please do not solve using facts about the index)
A relation ⪯ is defined on ℚ by ?⪯? if and only if ?=?+6? for some non-negative integer ?. Prove that ⪯ is a partial order.
2. Consider the relation E on Z defined by E n, m) n+ m is even} equivalence relation (a) Prove that E is an (b) Let n E Z. Find [n]. equivalence relation in [N, the equivalence class of 3. We defined a relation on sets A B. Prove that this relation is an (In this view, countable sets the natural numbers under this equivalence relation). exactly those that are are 2. Consider the relation E on Z defined by...