Test from Discrete mathematics 24/11/2016
Name and surname 1 2 3 4 5 Sum
Two points for every task. Use a space below the tasks for answers.
1. Find a rule defining some bijection f : N → S−
0 where S−
0 is a set of nonpositive
even integers.
2. Let R, S be relations on set {1, 2, 3}. Decide if the following implications
are valid. Prove your claim.
a) R ◦ S is transitive ⇒ R, S are transitive,
b) R, S are antisymmetric ⇒ R ∪ S is antisymmetric.
3. Find some mapping f : R → R, such that its kernel Jf satisfies
x Jf y ⇔ x = y ∨ x + y = 2.
4. List all partitions of set {1, 2, 3, 4} provided that [1] = [2].
5. For given relation α = {(a, b), (a, c)} on set {a, b, c, d} find a smallest
relation β which is an equivalence relation and α ⊆ β.