For each of the following relations on the set of people, determine whether it i

Question

For each of the following relations on the set of people, determine
whether it is an equivalence relation, a total ordering, or neither. Which
of the dening properties of an equivalence relation or a total ordering
does it satisfy? (he put random symbols in the question. I'm writing these symbols in parenthesis since they do not come up on here. The first four are the symbols for a deck of cards)
(a) A(clubs)B if A is smellier than B.
(b) A(diamond)B if A and B are the same age (to the nearest year).
(c) A(heart)B if A loves B.
(d) A(spade)B if A was born on one of the three days of the week following
the day of the week on which B was born.
(e) A(star)B if A is a banker
Please answer the question fully. thanks :)

Explanation / Answer

If X is totally ordered under =, then the following statements hold for all a, b and c in X: 1.If a = b and b = a then a = b (antisymmetry); 2.If a = b and b = c then a = c (transitivity); 3.a = b or b = a (totality). A given binary relation ~ on a set A is said to be an equivalence relation if and only if it is reflexive, symmetric and transitive. Equivalently, for all a, b and c in A: 4 a ~ a. (Reflexivity) 5 if a ~ b then b ~ a. (Symmetry) 6 if a ~ b and b ~ c then a ~ c. a) a total ordering b)a total ordering c)neither d)not transitive hence neither e)neither

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