Truth tables or Boolean algebra

A question is this type if and only if it involves constructing truth tables or using Boolean algebra to prove logical statements.

2 questions · Easy -1.1

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OCR MEI D2 2007 June Q1
16 marks Moderate -0.5
1
  1. A joke has it that army recruits used to be instructed: "If it moves, salute it. If it doesn't move, paint it." Assume that this instruction has been carried out completely in the local universe, so that everything that doesn't move has been painted.
    1. A recruit encounters something which is not painted. What should he do, and why?
    2. A recruit encounters something which is painted. Do we know what he or she should do? Justify your answer.
  2. Use a truth table to prove \(( ( ( m \Rightarrow s ) \wedge ( \sim m \Rightarrow p ) ) \wedge \sim p ) \Rightarrow s\).
  3. You are given the following two rules. $$\begin{aligned} & 1 \quad ( a \Rightarrow b ) \Leftrightarrow ( \sim b \Rightarrow \sim a ) \\ & 2 \quad ( x \wedge ( x \Rightarrow y ) ) \Rightarrow y \end{aligned}$$ Use Boolean algebra to prove that \(( ( ( m \Rightarrow s ) \wedge ( \sim m \Rightarrow p ) ) \wedge \sim p ) \Rightarrow s\).
OCR MEI D2 2010 June Q1
16 marks Easy -1.8
1
  1. Mickey ate the last of the cheese. Minnie was put out at this. Mickey's defence was "There wasn't enough left not to eat it all". Let "c" represent "there is enough cheese for two" and "e" represent "one person can eat all of the cheese".
    1. Which of the following best captures Mickey's argument? $$\mathrm { c } \Rightarrow \mathrm { e } \quad \mathrm { c } \Rightarrow \sim \mathrm { e } \quad \sim _ { \mathrm { c } } \Rightarrow \mathrm { e } \quad \sim _ { \mathrm { c } } \Rightarrow \sim \mathrm { e }$$ In the ensuing argument Minnie concedes that if there's a lot left then one should not eat it all, but argues that this is not an excuse for Mickey having eaten it all when there was not a lot left.
    2. Prove that Minnie is right by writing down a line of a truth table which shows that $$( c \Rightarrow \sim e ) \Leftrightarrow ( \sim c \Rightarrow e )$$ is false.
      (You may produce the whole table if you wish, but you need to indicate a specific line of the table.)
    1. Show that the following combinatorial circuit is modelling an implication statement. Say what that statement is, and prove that the circuit and the statement are equivalent.
      \includegraphics[max width=\textwidth, alt={}, center]{c3a528e4-b5fe-4bff-a77e-e3199bb225a1-2_188_533_1272_767}
    2. Express the following combinatorial circuit as an equivalent implication statement.
      \includegraphics[max width=\textwidth, alt={}, center]{c3a528e4-b5fe-4bff-a77e-e3199bb225a1-2_314_835_1599_616}
    3. Explain why \(( \sim \mathrm { p } \wedge \sim \mathrm { q } ) \Rightarrow \mathrm { r }\), together with \(\sim \mathrm { r }\) and \(\sim \mathrm { q }\), give p .