Math 199 Final PTL with
100% correct answers 2025
Prop 20 - answer Let ~ be an equivalence relation on X. Let a, b in
X. Then a ~ b iff [a] = [b]
Prop 21 - answer Let X be a set and ~ is an equivalence relation on
X. Then the quotient, X/~, will always be a partition of X.
Prop 22 - answer Let P be a partition on X. Then ~p is an
equivalence relation on X.
Prop 23 - answer Let P be a partition on the set X, then X/~p = P
Prop 24 - answer Let X be a set and ~ be an equivalence relation on
X. Let a, b in X. Then a~b iff a~(X/~)b.
Prop 25 - answer If X is a set whose elements are sets, then
cardinality is an equivalence relation on X. (ARB iff |A| = |B| is an ER)
Prop 26 - answer A set A is countable iff its elements can be listed in
an infinite sequence, possible with repetitions
Lemma from WA7 #4 - answer Let X be a set, P be a partition of X,
and x in X. then [x]~p = A for some A in P.
Theorem 27 - answer The set of rational numbers is countable
Lemma 28 - answer Q+ (the positive rationals) is countable
100% correct answers 2025
Prop 20 - answer Let ~ be an equivalence relation on X. Let a, b in
X. Then a ~ b iff [a] = [b]
Prop 21 - answer Let X be a set and ~ is an equivalence relation on
X. Then the quotient, X/~, will always be a partition of X.
Prop 22 - answer Let P be a partition on X. Then ~p is an
equivalence relation on X.
Prop 23 - answer Let P be a partition on the set X, then X/~p = P
Prop 24 - answer Let X be a set and ~ be an equivalence relation on
X. Let a, b in X. Then a~b iff a~(X/~)b.
Prop 25 - answer If X is a set whose elements are sets, then
cardinality is an equivalence relation on X. (ARB iff |A| = |B| is an ER)
Prop 26 - answer A set A is countable iff its elements can be listed in
an infinite sequence, possible with repetitions
Lemma from WA7 #4 - answer Let X be a set, P be a partition of X,
and x in X. then [x]~p = A for some A in P.
Theorem 27 - answer The set of rational numbers is countable
Lemma 28 - answer Q+ (the positive rationals) is countable
