(1)
a ¹~ b⇒
(a) a ¹~ a (reflexivity)
(b) a ¹~ b and b ¹~ a (symmetry)
(c) a ¹~ b and b ¹~ c⇒a ¹~ c (transitivity);
a ²~ b⇒
(a) a ²~ a (reflexivity)
(b) a ²~ b and b ²~ a (symmetry)
(c) a ²~ b and b ²~ c⇒a ²~ c (transitivity).
³~ = ¹~ ∩²~, ⁴~ = ¹~ ∪²~.
(a) a ³~ a⇒
(a ¹~ a)∩(a ²~ a)⇒TRUE∩TRUE=TRUE (reflexivity)
(b) a ³~ b and b ³~ a⇒
((a ¹~ b)∩(a ²~ b))∩((b ¹~ a)∩(b ²~ a))=
(TRUE∩TRUE)∩(TRUE∩TRUE)=TRUE∩TRUE=TRUE (symmetry)
(c) [a ³~ b and b ³~ c⇒a ³~ c]⇒
[((a ¹~ b)∩(a ²~ b))∩((b ¹~ c)∩(b ²~ c))⇒((a ¹~ c)∩(a ²~ c))];
[(TRUE∩TRUE)∩(TRUE∩TRUE)⇒(TRUE∩TRUE)]⇒
[(TRUE∩TRUE)⇒(TRUE∩TRUE)].
Since the statement and its implication are identical, there is no inconsistency, so ³~ is transitive. All three requirements have been met, so ³~ is an equivalence relation.
(a) a ⁴~ a⇒
(a ¹~ a)∪(a ²~ a)⇒TRUE∪TRUE=TRUE (reflexivity)
(b) a ⁴~ b and b ⁴~ a⇒
((a ¹~ b)∪(a ²~ b))∩((b ¹~ a)∪(b ²~ a))=
(TRUE∪TRUE)∩(TRUE∪TRUE)=TRUE∩TRUE=TRUE (symmetry)
(c) [a ⁴~ b and b ⁴~ c⇒a ⁴~ c]⇒
[((a ¹~ b)∪(a ²~ b))∩((b ¹~ c)∪(b ²~ c))⇒((a ¹~ c)∪(a ²~ c))];
[(TRUE∪TRUE)∩(TRUE∪TRUE)⇒(TRUE∩TRUE)]⇒
[(TRUE∩TRUE)⇒(TRUE∩TRUE)].
Since the statement and its implication are identical, there is no inconsistency, so ⁴~ is transitive. All three requirements have been met, so ⁴~ is an equivalence relation.
(2)
Let B={1,8,27,...,k³}, b=f(a)=a³.
b ᴮ~ b⇒[(b mod 9)∈{0,1}]=[(b mod 9)∈{0,1}] (reflexive).
When k=3n, k³=27n³ (∀n∈ℤ) b mod 9=0;
when k=3n±1, k³=(3n)³±3(3n)²+3(3n)±1=27n³±27n²+9n±1.
This can be written 9n(3n²±3n+1)±1 or 9P±1. (9P±1)mod 9=1 or 9-1=8. Hence ∀b∈B, ᴮ~ is (b mod 9)∈{0,1,8}.
b₁ ᴮ~ b₂ and b₂ ᴮ~ b₁ for (b₁,b₂)∈B, and for (b₁,b₂,b₃)∈B:
b₁ ᴮ~ b₂ and b₂ ᴮ~ b₃⇒b₁ ᴮ~ b₃ (symmetry and transitivity).
Let A={-k,...,-2,-1,0,1,2,3,...,k}. But ∀a∈A is not required to obey an equivalence relationship dependent on that in B, so ᴬ~ does not need to be an equivalence relation.
(3)
Let A be the set of all married women in a town and B be the set of all married man. ᴬ~ relates the elements of A (all married female) and ᴮ~ relates those of B (all married male). The product X=A×B is the set of married couples (a,b) ∀a∈A and ∀b∈B. But of all the possible pairs (a,b) only one represents a couple married to one another. So, if ˣ~ represents marriages, this equivalence relation cannot apply to all pairs in X. Marriages would be a subset of X and such an equivalence relation would apply.