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

Question

Given sets A = {a, b, c, d, e}, and B = {1, 3, 5}, answer the following: Show a smallest relation that is both reflexive and symmetric, _BR_B. Show a non-empty, partial function, _aR_B. Show a complete function, _aR_B, that does not cover B. Show a complete, 1-to-1 function _BR_a. Show a relation, _aR_A 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.(a) {(1,1)} is a smallest relation that is both reflexive and symmetric.

(b) {(a,1),(b,2),(c,3)} is a partial function.

(c) {(a,1),(b,2),(c,1),(d,2),(e,1)} is a complete function that does not cover B.

(d) {(1,a),(2,b),(3,c)} is a complete one-one function.

(e) {(a,b),(b,c),(c,d)} is not transitive. Because (a,b) and (b,c) are in the relation but (a,c) is not.

2. Let A be a rectangle

Area of A = Area of A

=> The relation is symmetric

Suppose Area of A = Area of B

Then Area of B = Area of A

=> The relation is symmetric

Suppose Area of A = Area of B and Area of B = Area of C

Then Area of C = Area of B = Area of A

=> The relation is transitive.

=> The relation is an equivalence relation.

3. No. The rectangles are equivalent only if the relation is area.

Consider a rectangle with area 12. It's sides could be 3*4. There could be another rectangle with area 12 but sides 6 and 2. If the relation is sum of sides, one would have 7 and the other 8. So they are not necessarily equivalent.

Related Questions

Navigate

• Browse (All)
• Browse G
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.