Be careful! (b) & (c) look alike but are not.
(b) is transtive but (c) is not. WHY?
As a rule of thumb: the word "same" implies an equivlance relation.
I am not sure how to go about doing the following questions... any help will be highly appreciated..
Q. 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 parent}
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}
MY Answers:
a) a and b are same age, therefore it is reflexive since a = b
It is symmetric a = b and b = a, it is transitive since a = b and b = c implies a = c.
b) a and b have same parent, therefore it is reflexive since a = b, symmetric since a = b and b = a, but NOT transitive since a = b and b = c, but a != c.
Similarly the rest of them... but I don't have a clue how to go about doing it...
Let’s change (b) to “have the same mother”.
It is easy to see reflexive and symmetric.
If a & b have the same mother and b & c have the same mother then surly a & c have the same mother. So it is transitive.
But have a parent in common? Here a & b may have the same mother and b & c have the same father, but a & c may not have a common parent. Part (e) is exactly the same idea because some people are bilingual. A may speak English, B may speak English and French and C may speak French.