Question -
Answer -
(i) Consider R1
Given R1┬а= {(a, a), (a, b), (a, c), (b, b), (b, c), (c, a), (c,b), (c, c)}
Now we have check R1┬аisreflexive, symmetric and transitive
Reflexive:
Given (a, a), (b, b)and (c, c) тИИ R1
So, R1┬аisreflexive.
Symmetric:
We see that the ordered pairs obtained by interchanging the components of R1┬аarealso in R1.
So, R1┬аis symmetric.
Transitive:
Here, (a,┬аb) тИИR1,┬а(b,┬аc)тИИR1┬аand┬аalso┬а(a,┬аc)тИИR1
So, R1┬аistransitive.
(ii) Consider R2
Given R2┬а={(a, a)}
Reflexive:┬а
Clearly┬а(a,a)┬атИИR2.
So, R2┬аisreflexive.
Symmetric:┬а
Clearly┬а(a,a)┬атИИR┬а
тЗТ┬а(a, a)┬атИИR.
So, R2┬аissymmetric.
Transitive:┬а
R2┬аisclearly a transitive relation, since there is only one element in it.
(iii) Consider R3
Given R3┬а= {(b, c)}
Reflexive:
Here,(b,┬аb)тИЙ┬аR3┬аneither┬а(c,┬аc)┬атИЙ┬аR3
So, R3┬аisnot reflexive.
Symmetric:
Here, (b,┬аc)┬атИИR3,┬аbut┬а(c,b)┬атИЙR3
So, R3┬аisnot symmetric.
Transitive:
Here, R3┬аhas only two elements.
Hence, R3┬аistransitive.
(iv) Consider R4
Given R4┬а= {(a, b), (b, c), (c, a)}.
Reflexive:
Here, (a,┬аa)┬атИЙ┬аR4,┬а(b,┬аb)тИЙ┬аR4┬а(c,┬аc)тИЙ┬аR4
So,┬аR4┬аis┬аnot┬аreflexive.
Symmetric:
Here, (a,┬аb) тИИ┬аR4,┬аbut┬а(b,a)┬атИЙ┬аR4.
So,┬аR4┬аis┬аnot┬аsymmetric
Transitive:
Here, (a,┬аb) тИИR4,┬а(b,┬аc)тИИR4,┬аbut┬а(a,┬аc)тИЙR4
So,┬аR4┬аis┬аnot┬аtransitive.