Question

For each of the following relations, determine whether the
relation is reflexive, symmetric, antisymmetric, or transitive.
a) R ⊆ Z+ X Z+ where a R b if a|b).
b) R is the relation on Z where a R b if a|b.
c) For a given universe U and a fixed subset C of U, define
R on P (U) as follows: For A, B ⊆ U we have A R B if
A ∩ C = B ∩ C.
d) On the set A of all lines in R2, define the relation _ for
two lines _1, _2 by _1 _ _2 if _1 is perpendicular to _2.
e) R is the relation on Z where x _ y if x + y is odd.
f ) R is the relation on Z where x _ y if x − y is even.

Answer 1

For each of the following relations, determine whether the
relation is reflexive, symmetric, antisymmetric, or transitive.
a) R ⊆ Z+ X Z+ where a R b if a|b).
b) R is the relation on Z where a R b if a|b.
c) For a given universe U and a fixed subset C of U, define
R on P (U) as follows: For A, B ⊆ U we have A R B if
A ∩ C = B ∩ C.
d) On the set A of all lines in R2, define the relation _ for
two lines _1, _2 by _1 _ _2 if _1 is perpendicular to _2.
e) R is the relation on Z where x _ y if x + y is odd.
f ) R is the relation on Z where x _ y if x − y is even.
g) Let T be the set of all triangles in R2. Define R on T by
t1 R t2 if t1 and t2 have an angle of the same measure.
h) R is the relation on Z X Z where (a, b)_(c, d) if a ≤ c.
[Note: R ⊆ (Z X Z) X(Z X Z).]

Answer 2

I actually don't know what antisymmetric means, but I guess I can help out with the other ones...
Let's start with (a)

Answer 3

Do you remember what 'reflexive' means?