Exercises
- Prove that
- for arbitrary sets A, B, C, A ×
(B\C) = (A × B)\ (A ×
C).
- if A, B, C are non-empty sets, then
A ⊆ B and C ⊆ D
iff A × C ⊆ B × D.
- for every A, B, C holds (A ⊕ B) × C = (A × C) ⊕ (B × C).
- Let U = {0,1,2,3} and let r1, r2 ⊆
U × U be binary relations such that
- (n,m) ∈ r1 iff m-n is even number,
- (n,m) ∈ r2 iff m ≤ n.
Describe each of these relations: as a set of ordered pairs, in the
form of a matrix and in the form of a graph. For each of them verify
the properties (symmetry, reflexivity, transitivity, etc.).
- Verify the properties of the relations:
- x r y iff 2|(x+y) for x,y ∈
N
- x r y iff sgn x ≤ sgn y x,y ∈ R
- x r y iff |x+y+1| ≤ 1 x,y ∈
R (draw the diagram of this relation)
- Give an example (as a set, as a matrix and as a graph) of the
binary relation in the set X={a,b,c,d} such that it is
- reflexive,
- symmetric,
- transitive.
- Let x0 be a fixed positive real number. Draw the diagram of the
relation (r1 ∪ r2) knowing that
r1 = {(x,y) ∈ R+ × R : y = - √
x and x ≤ x0} r2 = {(x,y) ∈ R+ ×
R : y = + √x and x >x0}
- Prove that
- If r1 and r2 are reflexive relations, then the relation (r1 ∩ r2) is reflexive too.
- Relation r is transitive iff r o r ⊆
r
- ( r1 ∪ r2) -1 = r1-1
∪ r2-1
- Determine the composition r o r, if r = {(x,y) ∈ R × R : x+y ≤ 0}.
- How many binary relations can one construct in the set of n
elements?
- Assuming that the relation is represented by the incidence
matrix, give the algorithms verifying reflexivity and irreflexivity of
it.