Mатематическая логика

Помочь написать контрольную работу. 

Пример прошлого года. Eсли понадобится, я могу прислать материалы.

Task 1 (35%)

a) Describe two major mistakes that you should not make, while writing formulas in the task (b).

b) Predicate language R has the following predicates:

Pr(x) (“x is a product”), Fa(x) (“x is a facrory”), Ci(x) (“x is a city”),

produces(x, y) (“[factory] x produces [product] y”),

located(x, y) (“[factory] x is located [in city] y”),

buys(x, y, z) (“[factory] x buys [product] y from [factory] z”), and also x=y.

Write down in the language R the following statements:

“each city is not a product and not a factory”;

“only factories can produce, and produce only products”;

“product x is produced at one and only one factory”;

“product y is produced only in one city”;

“factories of city x do not buy products, that are produced in city y”.

Task 2 (35%)

Write the sequences of formulas that prove:

a) [L1-L10, MP]: A→(B→C);A∧B ⊢ C;

b) [L1-L10, MP]: (D∧E)∧F ⊢ D∧(E∧F).

It is allowed to use only given hypotheses, axioms and MP. Keep in mind that it can be enough to use just some of axioms, none of the proofs will require to use all L1-L10.

Task 3 (15%)

Write the sequence of formulas that proves:

[L1-L2, MP]: A→(B→C);B ⊢ A→C.

It is allowed to use only given hypotheses, axioms and MP.

Task 4 (15%)

Write down at least 7 lines about the following: what do mean and why are necessary axioms L1 and L10.