{F} Construct a relation on the set {a, b, c, d} that is a. reflexive, symmetric, but not transitive. b. irreflexive, symmetric, and transitive. c. irreflexive, antisymmetric, and not transitive. d. reflexive, neither symmetric nor antisymmetric, and transitive. e. neither reflexive, irreflexive, symmetric, antisymmetric, nor transitive.
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Let us consider a set
Reflexivity: A relation R on set S is called reflexive if for all
Symmetry: A relation R on set S is called symmetric if whenever for all
Anti symmetry: A relation R on set S is called anti symmetric if and than for all
Transitivity: a relation R on set S is called transitive if and than for all
a) Consider a relation on such that
Now we can see that for all thus is reflexive.
Also whenever and whenever thus we can say is symmetric.
Again we can see that if and but Hence is not transitive.
b) Let . Consider on if b R R a then a R b " hence the R is symmetric.
Let , if c R a then a R b or b R R c " hence the R is transitivity.
Now take any element and observe that . Thus R is not reflexive, hence it is irreflexive.
c) Consider a relation on such that
We can see that so is not reflexive and also we can see that no element is related to each other means it is irreflexive.
Also we have if a=b whenever and but here so it is antisymmetric.
Again we can see that if and but Hence is not transitive.
d) Consider a relation on such that
We can see that for all thus is reflexive.
Also but thus we can say is not symmetric. Now whenever but thus is not antisymmetric.
Again we can see that if and then Hence is transitive.
e) Consider a relation on such that
We can see that so is not reflexive and also we can see that means it is not irreflexive.
Also but thus we can say is not symmetric. Now whenever but thus is not antisymmetric.
Again we can see that if and then Hence is transitive