1. Which of these relations on the set of all people are equivalence relations? Determine the properties of an equivalence relation that the others lack. a) {(a, b) | a and b are the same age}. b) {(a, b) | a and b have the same parents}.c) {(a, b) | a and b share a common parent}.d) {(a, b) | a and b have met}.e) {(a, b) | a and b speak a common language}. 2. Find equivalence classes of following relations if they exist. a) {(0, 0), (1, 1), (2, 2), (3, 3).b) {(0, 0), (0, 2), (2, 0), (2, 2), (2, 3), (3, 2), (3, 3)}.c) {(0, 0), (1, 1), (1, 2), (2, 1), (2, 2), (3, 3)}.d) {(0, 0), (1, 1), (1, 3), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}.e) {(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0), (2, 2), (3, 3)}.

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