## 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 \(f\Rightarrow e\).

(c) Determine whether your statement in part (a) is logically equivalent to your statement in part (b). Justify your answer.

## 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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