Test from Discrete mathematics 20/10/2016
Name and surname 1 2 3 4 5 Sum
Two points for every task. Use a space below the tasks for answers.
1. a) Decide whether formula ϕ = (∀x)(x = 0 → x = x + 1) is valid in R, Z,
resp. N, and explain why.
b) Write a negation of ϕ and modify it to a form in which the negation
operation may appear just with the subformulas without logical connectives
(atomic subformulas).
2. Write as a formula:
a) One can express b from equation a = b + 5 for any a, b.
b) 0 is not a smallest number.
3. Express the set A provided that ∅ ∈ A, A ⊆ {∅, {∅}}, A = {{∅}}.
4. For any sets A, B, C, prove
A × (B ∩ C) = (A × B) ∩ (A × C).
5. Express the sets as lists of elements:
a) P(P({∅})),
b) {{∅}, ∅} × {{∅}, ∅}.