Respuesta :
Answer:
Step-by-step explanation:
Equivalence relations are one which satisfy
i) Reflexive i.e. (x,x) belong to R for all x
ii) Symmetric if (x,y) is in R, then (y,x) also would be in R
iii) Transitive (x,y) and (y,z) imply (x,z)
a) {(a, b) | a and b are the same age}.
This is equivalence since all conditions are satisfied.
b) {(a, b) | a and b have the same parents}.
Equivalence since reflexive, symmetric and transitivity not necesary.
c) {(a, b) | a and b share a common parent}.
(a,b) and(b,c) need not imply (a,c) so not equivalent
(Because a and b have same mother while b and c have same father)
d) {(a, b) | a and b have met}.
Not transitive because (a,b) and (b,c) need not imply (a,c)
e) {(a, b) | a and b speak a common language}.
Not equivalence suppose a,b speak common English, b and c speak common french then a,c may not have common language.
2. Find equivalence classes of following relations if they exist.
a) {(0, 0), (1, 1), (2, 2), (3, 3). Equivalence.
b) {(0, 0), (0, 2), (2, 0), (2, 2), (2, 3), (3, 2), (3, 3)}.Equivalence
c) {(0, 0), (1, 1), (1, 2), (2, 1), (2, 2), (3, 3)}.Equivalence
d) {(0, 0), (1, 1), (1, 3), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}. Not transitive because (1,3) and (3,2) are there but (1,2) not there
e) {(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0), (2, 2), (3, 3)}.Equivalence
