Consider the binary relation R on the domain of four-letter strings de?ned by sR

Question

Consider the binary relation R on the domain of four-letter strings de?ned
by sRt if t is formed from the string s by cycling its characters one position left.
That is, abcdRbcda, where a, b, c, and d are individual letters. Determine whether
R is (a) re?exive, (b) symmetric, (c) transitive, (d) a partial order, and/or (e) an
equivalence relation. Give a brief argument why, or a counterexample, in each case.

Let S be the binary relation consisting of R applied 0 or more times. Thus, abcdSabcd,
abcdSbcda, abcdScdab, and abcdSdabc. Put another way, a string is related by S to
any of its rotations. Answer the five questions

Explanation / Answer

considering relation sRt,

a)-> checking for reflexive, reflexive propert says that, x R x

we can see here aaaa R aaaa, bbbb R bbbb, cccc R cccc, and dddd R dddd,

means set relates to itself,

Hence this relation is reflexive

b) -> symmetric property says that, if x R y, then y R x

now considering our case,

abcd R bcda nut we can see that bcda does not relate to abcd

so Given relation is not symmetric

c) -checking for transitive,

transitive property says that, if x R y and y R z then x R z,

considering our case,

abcd R bcda, and bcda R cdab but

abcd does not relate to cdab

so given relation is not transitive

d) -check for partial order,

R is a partial order relation if R is reflexive, antisymmetric and transitive.

but we have already seen that function is not transitive,

so this function is not partial order.

e) - for equivalence relation, R it must be reflexive, symmetric, and transitive

but R is neither symmetric or transitive in our case

so thi is not an equivalence relattion

ANSWER

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