Here are a couple more.
Code:
13.1 For each of the following relations defined on the set {1, 2, 3, 4, 5}, determine whether the relation is reflexive, irreflexive, symmetric, antisymmetric, and/or transitive.
a. R={(1,1),(2,2),(3,3),(4,4),(5,5)}.
b. R={(1,2),(2,3),(3,4),(4,5)}.
c. R={(1,1),(1,2),(1,3),(1,4),(1,5)}.
d. R={(1,1),(1,2),(2,1),(3,4),(4,3)}.
e. R={1,2,3,4,5}×{1,2,3,4,5}.
Where
a. is "reflexive, symmetric, antisymmetric, transitive", and
b. is "irreflexive, antisymmetric".
Code:
14.1 Which of the following are equivalence relations?
a. R={(1,1),(1,2),(2,1),(2,2),(3,3) } on the set {1, 2, 3}.
b. R={(1,2),(2,3),(3,1) } on the set {1, 2, 3}.
c. | on Z.
d. ≤ on Z.
e. {1,2,3}×{1,2,3} on the set {1,2,3}.
f. {1,2,3}×{1,2,3} on the set {1,2,3,4}.
g. Is-an-anagram-of on the set of English words. (For example, STOP is an anagram of POTS because we can form one from the other by rearranging its letters.)
Where
a. is "Yes",
f. is "No", and
g. is "Yes".
Code:
14.5 For each equivalence relation, find the requested equivalence class.
a. R={(1,1),(1,2),(2,1),(2,2),(3,3),(4,4)} on {1,2,3,4}. Find [1].
b. R={(1,1),(1,2),(2,1),(2,2),(3,3),(4,4)} on {1,2,3,4}. Find [4].
c. R is has-the-same-tens-digit-as on the set {x∈Z∶100<x<200}. Find [123].
d. R is has-the-same-parents-as on the set of all human beings. Find [you].
e. R is has-the-same-birthday-as on the set of all human beings. Find [you].
f. R is has-the-same-size-as on 2^({1,2,3,4,5}). Find [{1,3}].
Where
a. is "[1]={1, 2}", and
e. is "[you] is the set containing all people born on your birthday".
The given statements are hints and answers, given from the textbook.