(a) Does that mean a ~ b if and only if gcd(a;b) is even ? Because there is no relation written in (a), but just a subset of N².
In the case of a ~ b if and only if gcd(a;b) is even, then no, it is not a relation of equivalence, because not every a satisties : a ~ a.
In other words, that relation is not reflexive.
(b) Same guess : a ~ b if and only if cos (a) = cos (b).
This is a relation of equivalence, because :
a ~ a for every a,
if a ~ b, then b ~ a,
if a ~ b and b ~ c, then a ~ c.
The partition of the set of real numbers that arise from it is :
R = ∪ { θ + 2k.pi ; - θ + 2k.pi, k ∈ Z } , union with θ lying in [0;pi]
in other words, for every possible value of cos between -1 and 1, choose 1 real θ whose cos equals that value.
(c) ~ is not a relation of equivalence.
(x₁ ; y₁) ~ (x₂ ; y₂) simply means that the vectors are colinear.
This is not transitive :
any vector V is colinear to the nul vector, the nul vector is colinear to any vector W, but that doesn't mean that V and W are colinear.
(d) is a relation of equivalence.
The partition of R² is the union of concentric circles of radius R > 0, and the set containing only (0;0).