Question #287027

{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.


1
Expert's answer
2022-02-02T16:33:51-0500

Let us consider a set {a,b,c,d}.\{a, b, c, d\}.

Reflexivity: A relation R on set S is called reflexive if (a,a)R(a, a) \in R for all aS.a \in S.

Symmetry: A relation R on set S is called symmetric if (b,a)R(b, a) \in R whenever (a,b)R(a, b) \in R for all a,bS.a, b \in S.

 

Anti symmetry: A relation R on set S is called anti symmetric if (b,a)R(b, a) \in R and (a,b)R(a, b) \in R than a=ba=b for all a,bS.a, b \in S.

Transitivity: a relation R on set S is called transitive if (a,b)R(a, b) \in R and (b,c)R(b, c) \in R than (a,c)R(a, c) \in R for all a,b,cS.a, b, c \in S.

a) Consider a relation on {a,b,c,d}\{a, b, c, d\} such that

 

R1={(a,a),(b,b),(c,c),(d,d),(a,b),(b,a),(b,c),(c,b)}R_{1}=\{(a, a),(b, b),(c, c),(d, d),(a, b),(b, a),(b, c),(c, b)\}

 

Now we can see that for all x{a,b,c,d},(x,x)R1x \in\{a, b, c, d\},(x, x) \in R_{1} thus R1R_{1} is reflexive.

Also (b,a)R1(b, a) \in R_{1} whenever (a,b)R1(a, b) \in R_{1} and (c,b)R1(c, b) \in R_{1} whenever (b,c)R1(b, c) \in R_{1} thus we can say R1R_{1} is symmetric.

Again we can see that if (a,b)R1(a, b) \in R_{1} and (b,c)R1(b, c) \in R_{1} but (a,c)R1(a, c) \notin R_{1} Hence R1R_{1} is not transitive.

b) Let R2=R_{2}=\varnothing . Consider on a,b{a,b,c,d},a, b \in\{a, b, c, d\}, if b R R a then a R b " hence the R is symmetric.

Let a,b,c{a,b,c,d}a, b, c \in\{a, b, c, d\} , if c R a then a R b or b R R c " hence the R is transitivity.

Now take any element aSa \in S and observe that aR~aa \tilde{R} a . Thus R is not reflexive, hence it is irreflexive.

c) Consider a relation on {a,b,c,d}\{a, b, c, d\} such that

 

R3={(a,b),(b,c)}R_{3}=\{(a, b),(b, c)\}

 

We can see that (a,a)R3(a, a) \notin R_{3} so R3R_{3} 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 (a,b)R3(a, b) \in R_{3} and (b,a)R3(b, a) \in R_{3} but here (b,a)R3(b, a) \notin R_{3} so it is antisymmetric.

Again we can see that if (a,b)R1(a, b) \in R_{1} and (b,c)R1(b, c) \in R_{1} but (a,c)R1(a, c) \notin R_{1} Hence R3R_{3} is not transitive.

d) Consider a relation on {a,b,c,d}\{a, b, c, d\} such that

 

R4={(a,a),(b,b),(c,c),(d,d),(a,b),(b,a),(c,a),(b,c)}R_{4}=\{(a, a),(b, b),(c, c),(d, d),(a, b),(b, a),(c, a),(b, c)\}

 

We can see that for all x{a,b,c,d},(x,x)R4x \in\{a, b, c, d\},(x, x) \in R_{4} thus R4R_{4} is reflexive.

Also (b,c)R4(b, c) \in R_{4} but (c,b)R4(c, b) \notin R_{4} thus we can say R4R_{4} is not symmetric. Now (b,a)R4(b, a) \in R_{4} whenever (a,b)R4(a, b) \in R_{4} but aba \neq b thus R4R_{4} is not antisymmetric.

Again we can see that if (a,b)R4(a, b) \in R_{4} and (b,c)R4(b, c) \in R_{4} then (a,c)R4.(a, c) \in R_{4}. Hence R4R_{4} is transitive.

e) Consider a relation on {a,b,c,d}\{a, b, c, d\} such that

 

R5={(a,b),(b,a),(c,c),(a,c)}R_{5}=\{(a, b),(b, a),(c, c),(a, c)\}

 

We can see that (a,a)R5(a, a) \notin R_{5} so R5R_{5} is not reflexive and also we can see that (c,c)R5(c, c) \in R_{5} means it is not irreflexive.

 

Also (a,c)R5(a, c) \in R_{5} but (c,a)R5(c, a) \notin R_{5} thus we can say R5R_{5} is not symmetric. Now (b,a)R5(b, a) \in R_{5} whenever (a,b)R5(a, b) \in R_{5} but aba \neq b thus R5R_{5} is not antisymmetric.

Again we can see that if (b,a)R5(b, a) \in R_{5} and (a,c)R5(a, c) \in R_{5} then (b,c)R5.(b, c) \notin R_{5}. Hence R5R_{5} is transitive


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!
LATEST TUTORIALS
APPROVED BY CLIENTS