Investigation: Can you believe it?

Proofs Workshop

An election is taking place in the US. It is a twoâ&#x2C6;&#x2019;horse race between the Democrats and the Republicans, and no matter who wins, taxes will inevitably change after the election. A US politician tells you: i) If the Democrats win, income tax will rise ii) If the Republicans win, sales tax will fall 1) Assume, for the moment, that both these statements are true. Which of the following must be true? a) If income tax rises, then sales tax will rise b) If income tax falls, then sales tax will fall c) If sales tax rises, then income tax will rise d) If sales tax falls, then income tax will fall 2) For each of the following outcomes, decide whether the politician lied or not: a) The Republicans lost and income tax fell b) Income tax fell and sales tax rose c) The Republicans won and income tax rose d) Sales tax rose and the Democrats lost

Shirleen Stibbe

http://www.shirleenstibbe.co.uk

Proofs Workshop

An election is taking place in the US. It is a two−horse race between the Democrats and the Republicans, and whoever wins, taxes will inevitably change after the election. We define the following: • p: the Democrats win • q: income tax rises • r: sales tax falls A US politician tells you: i) If the Democrats win, income tax will rise ii) If the Republicans win, sales tax will fall Taking contrapositive of ii) and combining with i) gives: and its contrapositive:

p⇒ q ¬p ⇒ r ¬r ⇒ q ¬q ⇒ r

False if: p ∧ ¬q ¬p ∧ ¬r ¬q ∧ ¬r "

1) Assume, for the moment, that both these statements are true. Which of the following must be true? a) b) c) d)

If income tax rises, then sales tax will rise If income tax falls, then sales tax will fall If sales tax rises, then income tax will rise If sales tax falls, then income tax will fall

q ⇒ ¬r ¬q ⇒ r ¬r ⇒ q r ⇒ ¬q

   

2) For each of the following outcomes, decide whether the politician lied or not: a) b) c) d)

The Republicans lost and income tax fell Income tax fell and sales tax rose The Republicans won and income tax rose Sales tax rose and the Democrats lost

p ∧ ¬q, so he lied ¬q ∧ ¬r, so he lied ¬p ∧ q, he didn't lie ¬p ∧ ¬r, so he lied

Moral: US politicians lie only 75% of the time

Shirleen Stibbe

http://www.shirleenstibbe.co.uk

Investigation; Can you believe it

1) Assume, for the moment, that both these statements are true. Which of the following must be true? a) If income tax rises, then sales tax...