Question

probelms 9.1

9 Modular arithmetic Definition 9.1 Let S be a set. A relation R = R(,y) on S is a statement about pairs (x,y) of elements of S. For r,y ES, I is related to y notation: Ry) if R(x,y) is true. A relation Ris: Reflexive if for any I ES, R. Symmetric if for any ry ES, Ry implies y Rr. Transitive if for any r.y.ES, Ry and yRimply R. An equivalence relation is a reflexive, symmetric and transitive relation. Problem 9.1 Give 3 examples of nonequivalence relations. Definition 9.2 Let R be an equivalence relation on a set S and let reS. The equivalence class of ris ] = {y e SyRx}. Problem 9.2 Give 3 examples of equivalence relations and describe the equivalence classes. Problem 9.3 Let R be an equivalence relation on a set S. Prove that two equivalence classes are either equal or do not intersect. Conclude that S is a disjoint union of all equivalence classes Definition 9.3 Let a, b e Z and n E N a is congruent to b modulo n if (a - b): n. (Notation: a = b mod n.) Problem 9.4 Fix n E N. Prove that congruence modulo n is an equivalence relation on Z. How many equivalence classes does it have? Problem 9.5 Fix n E N. Prove that if a = b mod n and c ad mod n then a+c=b+d mod n and a-cab-d mod n. Problem 9.6 Fix n E N. Prove that if a = b mod n and c ad mod n then ac = bd mod n. Definition 9.4 Let n E N. Consider the equivalence relation congruence modulo n". Let Zin denote the set of all its equivalence classes, and for ae Z, let a. denote the equivalence class of a. Define the operations of addition, subtraction and multiplication on Zin by [al. + l. = (a + b). (a).-6. = (a - b). a.) = (ab] Problem 9.7 Does the above definition make sense? Why? Problem 9.8 Find the remainder of the division of 6220 +82019 by 7.

Answer 1

I 9.17 A relation will be non-equivalence if it bails to be either reflexive or symmetric or transitive. Ex. 17 let P be the set of people. R be a reln. on P defined by nRy = H is father of y.. clearly this is not reblerive as a can't be its own bather. . non-equivalence. Ex-2 Let R be the set of real numbers. R=< be the relu on R less than) il. HRY HLY It is also not reflexive. non-equivalence.

Ex. 3 Consider the set A={0, 1, 2}. R={(0,0), (1,1), (2, 2), (1,0), (1, 2)} Now, (1,0)ER but (0,1)&R So, R is not symmetric R is non-equivalence relation.
hope you like the answer ?

Any doubt feel free to comment ?