# Exam-Style Questions on Logic

## Problems on Logic adapted from questions set in previous Mathematics exams.

### 1.

IB Studies

At Valley Park school, students must choose at least one of three after-school activities: Sport, Music or Adventure.

Consider the following propositions about one of the students called Raj:

$$s$$: Raj chooses Sport,

$$m$$: Raj chooses Music,

$$a$$: Raj chooses Adventure.

(a) Write, in words, the compound proposition

$$\lnot s \Rightarrow (m \lor a)$$

(b) Complete the truth table for $$\lnot m \Rightarrow s$$

 $$m$$ $$s$$ $$\lnot m$$ $$\lnot m \Rightarrow s$$ T T T F F T F F

(c) State whether $$\lnot m \Rightarrow s$$ is a tautology, a contradiction or neither. Justify your answer.

### 2.

IB Studies

Two propositions $$e$$ and $$f$$ are defined as follows

$$e$$: Nigel is exercising regularly,

$$f$$: Nigel is getting fitter.

(a) Write down the following statement in words.

$$f\Rightarrow e$$

(b) Write down, in words, the contrapositive statement of $$f\Rightarrow e$$.

### 3.

IB Studies

Consider the following statements.

$$p$$: the actor was cast for a part in a play
$$q$$: the actor has learned all of her lines
$$r$$: the actor is ready for the dress rehearsal

(a) Write the following argument in symbolic form.

"If the actor was cast for a part in a play and has learned all of her lines, then the actor is ready for the dress rehearsal."

(b) Complete a truth table for the argument in part (a). Begin your truth table as follows.

 $$p$$ $$q$$ $$r$$ T T T T T F T F T T F F F T T F T F F F T F F F

(c) Use your truth table to determine whether the argument in part (a) is valid. Give a reason for your decision.

Write down the inverse of the argument in part (a)

(d) in symbolic form;

(e) in words.

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