### II.9 Exercises

#### Exercises [Slide 42]

Exercise 1.

Why are the following not propositional formulae? There might be more than one reason

• P¬Q
• P Q
• P¬ (Q )

Exercise 2.

As seen in the course notes   false,f1 f2,f1 f2 and f1 f2 are derived propositional formulae.

Write out ((P Q) (Q P)) using only ¬and .

Exercise 3.

Give the truth table for the following propositional formulae:

 false (¬P) ∨ (Q ∧ R) (¬P) ≡ (Q ∨¬R) (¬P) ≡ (Q ⊃ R)

Exercise 4.

Let σ0(P) = tt and σ0(Q) = tt.

Give the semantics of P Q, i.e., calculate MP Q(σ0).

Exercise 5.

Show that for any state σ0 and for propositional variables P and Q the following holds MP Q(σ0) = (MP(σ0) or MQ(σ0)).

Exercise 6.

Let P, Q, and R be propositional variables capturing the following propositions:

• P: You get a ﬁrst on the ﬁnal exam
• Q: You do every exercise of the course notes
• R: You get a ﬁrst for this module

Write the following as formulae using P, Q, and R and logical connectives.

• You get a ﬁrst for this module, but you do not do every exercise of the course notes.
• To get a ﬁrst for this module, it is necessary for you to get a ﬁrst on the ﬁnal exam.
• Getting a ﬁrst on the ﬁnal exam and doing every exercise in the course notes is suﬃcient for getting a ﬁrst in the module.

Exercise 7.

Determine which of the following formula is satisﬁable or valid. Explain why.

• false
• false true
• (P Q) ≡¬(¬P ¬Q)
• (P (P Q)) P

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