Given sets A = {a, b, c, d, e}, and B = {1, 3, 5}, answer the following: (a) Sho

Question

Given sets A = {a, b, c, d, e}, and B = {1, 3, 5}, answer the following: (a) Show a smallest relation that is both reflexive and symmetric, B^R_B. (b) Show a non-empty, partial function, A^R_B. (c) Show a complete function, A^R_B, that does not cover B. (d) Show a complete, 1-to-1 function B^RA. (e) Show a relation, A^RA, that is not transitive, and show why it is not transitive. Let R be the set of all rectangles. Show that the area of a rectangle imposes an equivalence relation on R. Does the previous problem show that all rectangles having the same area are equivalent?

Explanation / Answer

1) Gioven A={a,b,c,d,e} B ={1,3,5}

a) we know that the identity relation is reflexive as well as symmetric and transitive and also it is the smallest among others

so the smallest relation such that BRB which is symmetric as well as refelxive is identity relation

{(1,1),(3,3),(5,5)}

b) non empty partial function we can generate so many such partial functions from the given sets

one such example {(a,1),(b,3)}

such that no two first coordinates of ordered pairs should not repeated.

c) complete function is the ARB but doesnot cover B

that is codomain is not equal to range

one such example {(a, 1), (b,3),(c,1),(d,3),(e,5)}

d) complet one to one function BRA ( second coordinates also should not repeated to say it is one one)

{(1,a),(3,b),(5,c)}

e) a relation ARA which is not transtive

R= {(1,3)(3,1)}

clearly (a,b) belongs to R, (b,c) belongs to R but (a,c) doesnot belongs to R

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