Logical statements and converses

3 The converse of the statement ' \(P \Rightarrow Q\) ' is ' \(Q \Rightarrow P\) '.
Write down the converse of the following statement.
' \(n\) is an odd integer \(\Rightarrow 2 n\) is an even integer.'
Show that this converse is false.

4 Given that \(n\) is a positive integer, write down whether the following statements are always true (T), always false (F) or could be either true or false (E).

1. \(2 n + 1\) is an odd integer
2. \(3 n + 1\) is an even integer
3. \(n\) is odd \(\Rightarrow n ^ { 2 }\) is odd
4. \(n ^ { 2 }\) is odd \(\Rightarrow n ^ { 3 }\) is even

1 Explain why each of the following statements is false. State in each case which of the symbols ⟹, ⟸ or ⇔ would make the statement true.

1. ABCD is a square ⇔ the diagonals of quadrilateral ABCD intersect at \(90 ^ { \circ }\)
2. \(x ^ { 2 }\) is an integer \(\Rightarrow x\) is an integer

2 Complete each of the following by putting the best connecting symbol ⟵, ⟸ or ⇒) in the box. Explain your choice, giving full reasons.

1. \(n ^ { 3 } + 1\) is an odd integer □ \(n\) is an even integer
2. \(( x - 3 ) ( x - 2 ) > 0\) □ \(x > 3\)

5 Given that \(n\) is a positive integer, write down whether the following statements are always true (T), always false (F) or could be either true or false (E).

1. \(2 n + 1\) is an odd integer
2. \(3 n + 1\) is an even integer
3. \(n\) is odd \(\Rightarrow n ^ { 2 }\) is odd
4. \(n ^ { 2 }\) is odd \(\Rightarrow n ^ { 3 }\) is even

6 The converse of the statement ' \(\mathrm { P } \Rightarrow \mathrm { Q }\) ' is ' \(\mathrm { Q } \Rightarrow P\) '.
Write down the converse of the following statement. $$\text { ' } n \text { is an odd integer } \Rightarrow 2 n \text { is an even integer.' }$$ Show that this converse is false.

1 Is the following statement true or false? Justify your answer. $$x ^ { 2 } = 4 \text { if and only if } x = 2$$

1

1. When marking coursework, a teacher has to complete a form which includes the following:
   □
   In your opinion is this the original work of the pupil? (tick as appropriate)
   I have no reason to believe that it is not □
   I cannot confirm that it is □
   1. The teacher suspects that a pupil has copied work from the internet. For each box, state whether the teacher should tick the box or not.
   2. The teacher has no suspicions about the work of another pupil, and has no information about how the work was produced. Which boxes should she tick?
   3. Explain why the teacher must always tick at least one box.
2. Angus, the ski instructor, says that the class will have to have lunch in Italy tomorrow if it is foggy or if the top ski lift is not working. On the next morning Chloe, one of Angus's students, says that it is not foggy, so they can have lunch in Switzerland. Produce a line of a truth table which shows that Chloe's deduction is incorrect. You may produce a complete truth table if you wish, but you must indicate a row which shows that Chloe's deduction is incorrect.
3. You are given that the following two statements are true. $$\begin{aligned} & ( \mathrm { X } \vee \sim \mathrm { Y } ) \Rightarrow \mathrm { Z } \\ & \sim \mathrm { Z } \end{aligned}$$ Use Boolean algebra to show that Y is true.

2

1. Emelia: 'I won't go out for a walk if it's not dry or not warm.'
   Gemma: 'It's warm. Let's go!'
   Will what Gemma has said convince Emelia, and if not, why not?
2. If it is daytime and the car headlights are on, then it is raining. If the dashboard lights are dimmed then the car headlights are on.
   It is daytime.
   It is not raining.
   1. What can you deduce?
   2. Prove your deduction.
3. In this part of the question the switch X is represented by \includegraphics[max width=\textwidth, alt={}, center]{d254fbd2-7443-4b6d-87ba-f0d71fce5e17-3_104_138_824_1226} The switch can be wired into a circuit so that current flows when
   the switch is up \includegraphics[max width=\textwidth, alt={}, center]{d254fbd2-7443-4b6d-87ba-f0d71fce5e17-3_103_177_1005_593}
   but does not flow when it is down \includegraphics[max width=\textwidth, alt={}, center]{d254fbd2-7443-4b6d-87ba-f0d71fce5e17-3_111_167_1000_1334} Or the switch can be wired so that current flows when
   the switch is down \includegraphics[max width=\textwidth, alt={}, center]{d254fbd2-7443-4b6d-87ba-f0d71fce5e17-3_109_172_1228_639}
   but does not flow when it is up \includegraphics[max width=\textwidth, alt={}, center]{d254fbd2-7443-4b6d-87ba-f0d71fce5e17-3_109_174_1228_1327}
   1. Explain how the following circuit models \(( \mathrm { A } \wedge \mathrm { B } ) \Rightarrow \mathrm { C }\). \includegraphics[max width=\textwidth, alt={}, center]{d254fbd2-7443-4b6d-87ba-f0d71fce5e17-3_365_682_1484_694} In the following circuit B1 and B2 represent 'ganged' switches. This means that the two switches are either both up or both down. \includegraphics[max width=\textwidth, alt={}, center]{d254fbd2-7443-4b6d-87ba-f0d71fce5e17-3_364_1278_2042_397}
   2. Given that A is down, C is up and current is flowing, what can you deduce?

In each case, choose one of the statements $$P \Rightarrow Q \quad\quad P \Leftarrow Q \quad\quad P \Leftrightarrow Q$$ to describe the complete relationship between P and Q.

1. For \(n\) an integer: P: \(n\) is an even number Q: \(n\) is a multiple of 4 [1]
2. For triangle ABC: P: B is a right-angle Q: \(AB^2 + BC^2 = AC^2\) [1]

In each of the following cases choose one of the statements $$\text{P} \Rightarrow \text{Q} \qquad \text{P} \Leftrightarrow \text{Q} \qquad \text{P} \Leftarrow \text{Q}$$ to describe the complete relationship between P and Q.

1. P: \(x^2 + x - 2 = 0\) Q: \(x = 1\) [1]
2. P: \(y^3 > 1\) Q: \(y > 1\) [1]

Which of these statements is correct? Tick one box. [1 mark] \(x = 2 \Rightarrow x^2 = 4\) \(x^2 = 4 \Rightarrow x = 2\) \(x^2 = 4 \Leftrightarrow x = 2\) \(x^2 = 4 \Rightarrow x = -2\)