Let X be the set of all fourbit strings eg 0011 0101 1011 et

Let X be the set of all four-bit strings, e.g., 0011, 0101, 1011, etc. Define a relation R on X as (s_1, s_2) R if some substring of s_1 of length 2 is the equal to some substring of s_2 of length 2. Examples (0111, 1010) R since 01 is a substring of 0111 and 1010. But (1110, 0001) R because they do not possess a common substring of length 2. Is thus relation reflexive, symmetric, anti-symmetric, transitive and/or a partial order?

Solution

The relation is reflexive since (a,a) will be belonging to R, since (s1,s1) will have all many pairs of length which will match with each other

Hence the relation is reflexive in nature

The relation is said to be symmetric in R, if (s1,s2) belongs to R, then (s2,s1) belongs to R

Since (s1,s2) has two length string in common, then (s2,s1) will also satisfy the condition

Hence the relation is symmetric as well

The relation is said to be transitive, if (a,b), (b,c) belongs to R, then (a.c) belongs to R

relation will not necessary be a transitive relation

The relation is not antisymmetric since the relation is symmetric

The relation is not a poset since the relation is symmetric in nature