Question

Hide Assignment Information Instructions Let D={0,1}6 Determine if the following relations are equivalence relations or not. Justify your answers. (a) Define the relation R on domain D as: xRy iff y can be obtained from x by swapping any two bits. (b) Define the relation R on domain D as: xRy if y can be obtained from x by reordering the bits in any way.

Answer 1

The first relation, where bits are swapped, is not an equivalence relation as it may not be reflexive. The second relation, involving any reordering of bits, is an equivalence relation because it meets all necessary conditions.

Step-by-step explanation:

The student is asking whether certain relations on a specific set D = {0,1}6 are equivalence relations. An equivalence relation must satisfy three conditions: it must be reflexive, symmetric, and transitive.

(a) For the relation xRy where 'y' can be obtained from 'x' by swapping any two bits, this is not an equivalence relation. While it appears to be symmetric and transitive, it fails the reflexive test because there may be strings that cannot be obtained from themselves by swapping two different bits since the same bit cannot occupy two places at once.

(b) For the relation xRy where 'y' can be obtained from 'x' by reordering the bits in any way, this relation is an equivalence relation. Any string can be reordered to itself (reflexive), any reordering can be reversed (symmetric), and the result of one reordering can be further reordered (transitive).