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I started to learn discrete mathematics using "Discrete mathematics and its Application" book. In this book exercises I couldn't find answer to the below question.

Let

  • $p$: I bought a lottery ticket this week.

  • $q$: I won the million dollar jackpot on Friday.

Translate $$\neg p \lor (p \land q)$$ to English.

Community
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Blasanka
  • 151

2 Answers2

3

Let us do it in steps.

First, we just replace the symbols by their 'word' analogues.

NOT 'I bought a lottery ticket this week' OR ('I bought a lottery ticket this week' AND 'I won the million dollar jackpot on Friday.')

Second, let us smooth the sentence.

I did not buy a lottery ticket this week, or I bought a lottery ticket this week and I won the million dollar jackpot on Friday.

Third, let us think about what the sentence means.

There are two options 'I did not buy' or 'I did buy'; in the second case 'I won.' This means if I did buy, then I did win. So:

If I bought a lottery ticket this week, then I won the million dollar jackpot on Friday

Generally $\neg p \lor (p \land q)$ expresses "If p, then q."

quid
  • 42,135
2

Direct translation:

Either I did not buy a lottery ticket this week, or I bought a lottery ticket this week and I won the million dollar jackpot on Friday.

You can simplify this by relying on the logical equivalence of $\color\red{\neg{x}\vee{y}}$ and $\color\green{{x}\implies{y}}$.

Thus $\color\red{\neg{p}\vee({p}\wedge{q})}$ is equivalent to $\color\green{{p}\implies{p}\wedge{q}}$, which is quivalent to ${p}\implies{q}$.

Hence your statement can be translated to:

If I bought a lottery ticket this week, then I won the million dollar jackpot on Friday.

barak manos
  • 43,109