Exercises

1. Symbolize the following statements by formulas.

a) A relation is an equivalence relation if and only if it is reflexive, symmetric and transitive.

b) If the humidity is so high, it will rain either this afternoon or this evening.

c) Cancer will not be cured unless its cause is determined and a new drug for cancer is found.

d) It requires courage and skills to climb that mountain.

e) If he is a man, who campaigns so hard, he probably will be elected.

2. Given:

P = "he needs a doctor"

Q = "He needs a lawyer"

R = "He has an accident"

S = "He is sick"

U = "He is injured"

State the following formulas in English:

a) (S → P)∧(R→Q)

b) P→(S∨U)

c) (P∧Q)→R

d) (P∧Q)↔(S∧U)

e) ~(S∨U)→~P

3. For each of the following formulas, determine whether it is valid, invalid, inconsistent or some combination of these.

4. Consider the following statement: If the congress refuses to enact new laws, then the strike will not be over unless it lasts more than one year and the president of the firm resigns, and if either the congress enacts new laws or the strike is not over then the strike lasts more than one year.

Is the above statement contradictory? Explain.

5. Transform the following into disjunctive normal forms.

(S → P)∧(R→Q)

6. Transform the following into conjunctive normal forms.

(S → P)∧(R→Q)

7. Is it possible to have a formula that is in conjunctive normal form as well as disjunctive normal form? If yes, give an example.

8. Verify each of the following pairs of equivalent formulas by transforming them on both sides of the sign "=" into the same normal form.

(формулы см. в русс. книге)

9. Prove that(~Q →~P) is a logical consequence of (Q →P)

10. Consider the following statements:

F₁ = "Tom cannot be a good student unless he is smart and his father supports him."

F₂ = "Tom is a good student only if his father supports him."

Show that F₂ is a logical consequence of F₁.

11. Show that for the following statements, F₂ is a logical consequence of F₁.

F₁ = "If the president does not have the appropriate authority or if he does not want to take responsibility, then neither order will be restored nor will the riots stop spreading unless the rioters become tired of rioting and the local authorities begin to take conciliatory actions."

F₂ = "If the president does not want to take responsibility and the rioters are tired of rioting, then the riots will spread."

12. Show that Q is a logical consequence of (P→Q) and P. This is related to so-called modus ponens rule.