1.01b Logical connectives: congruence, if-then, if and only if

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

8 Fig. 8 shows a right-angled triangle with base \(2 x + 1\), height \(h\) and hypotenuse \(3 x\). \begin{figure}[h] \end{figure} Fig. 8

1. Show that \(h ^ { 2 } = 5 x ^ { 2 } - 4 x - 1\).
2. Given that \(h = \sqrt { 7 }\), find the value of \(x\), giving your answer in surd form.

1

1. Statement P is \(a + b = 4\).
   Statement Q is \(\quad a = 1\) and \(b = 3\).
   Which one of the following is correct? $$\mathrm { P } \Rightarrow \mathrm { Q } , \quad \mathrm { P } \Leftrightarrow \mathrm { Q } , \quad \mathrm { P } \Leftarrow \mathrm { Q }$$
2. Statement R is \(\quad x = 2\). Statement S is \(\quad x ^ { 2 } = 4\). Which one of the following is correct? $$R \Rightarrow S , \quad R \Leftrightarrow S , \quad R \Leftarrow S$$

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. \(( x - 3 ) ( x - 2 ) > 0\) □ \(x > 3\)

3 Select the best statement from $$\begin{aligned} & \mathrm { P } \Rightarrow \mathrm { Q } \\ & \mathrm { P } \Leftarrow \mathrm { Q } \\ & \mathrm { P } \Leftrightarrow \mathrm { Q } \end{aligned}$$ none of the above
to describe the relationship between P and Q in each of the following cases.

1. P: WXYZ is a quadrilateral with 4 equal sides \(\mathrm { Q } : \mathrm { WXYZ }\) is a square
2. P: \(n\) is an odd integer Q : \(\quad ( n + 1 ) ^ { 2 }\) is an odd integer
3. P : \(n\) is greater than 1 and \(n\) is a prime number Q : \(\sqrt { n }\) is not an integer

4 Show that the following statement is false. $$x - 5 = 0 \Leftrightarrow x ^ { 2 } = 25$$

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.

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

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

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

10 Select the best statement from $$\begin{aligned} & \mathbf { P } \Rightarrow \mathbf { Q } \\ & \mathbf { P } \Leftarrow \mathbf { Q } \\ & \mathbf { P } \Leftrightarrow \mathbf { Q } \end{aligned}$$ none of the above
to describe the relationship between P and Q in each of the following cases.

1. P : WXYZ is a quadrilateral with 4 equal sides \(\mathrm { Q } : \mathrm { WXYZ }\) is a square
2. P: \(n\) is an odd integer Q : \(\quad ( n + 1 ) ^ { 2 }\) is an odd integer
3. P: \(n\) is greater than 1 and \(n\) is a prime number Q : \(\sqrt { n }\) is not an integer

9 Complete each of the following by putting the best connecting symbol ( \(\Leftrightarrow , \Leftarrow\) or ⇒ ) in the box. Explain your choice, giving full reasons.

1. \(n ^ { 3 } + 1\) is an odd integer □ \(n\) is an even integer
2. \(( x - 3 ) ( x - 2 ) > 0\) □ \(x > 3\) Section B (36 marks)

1

1. The following was said in a charity appeal on Radio 4 in October 2006.
   "It is hard to underestimate the effect that your contribution will make."
   Rewrite the comment more simply in your own words without changing its meaning.
2. A machine has three components, A, B and C, each of which is either active or inactive.
   The states (active or inactive) of the components and the machine are to be modelled by a combinatorial circuit in which "active" is represented by "true" and "inactive" is represented by "false". Draw such a circuit.
3. Construct a truth table to show the following. $$[ ( ( \mathrm { a } \wedge \mathrm {~b} ) \vee ( ( \sim \mathrm { a } ) \wedge \mathrm { c } ) ) \vee ( ( \sim \mathrm { b } ) \wedge \mathrm { c } ) ] \Leftrightarrow \sim [ ( ( \sim \mathrm { a } ) \wedge ( \sim \mathrm { c } ) ) \vee ( ( \sim \mathrm { b } ) \wedge ( \sim \mathrm { c } ) ) ]$$

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.)
2. 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 .

1

1. Heard in Parliament: "Will the minister not now discontinue her proposal to ban the protest?"
   The minister replied "Yes I will."
   To what had the minister committed herself logically, and why might that not have been her intention?
2. In a cricket tournament an umpire might be required to decide whether or not a batsman is out 'lbw', ie 'leg before wicket'. The lbw law for the tournament refers to parts of the cricket pitch as shown in the diagram (assuming a right-handed batsman): \includegraphics[max width=\textwidth, alt={}, center]{52b8153f-e655-4852-a0f8-6f1c1e9c9170-2_254_1045_717_507} The umpire has to make a number of judgements:
   A Would the ball have hit the wicket?
   B Did the ball hit the batsman, or part of his equipment other than the bat, without hitting the bat?
   C Did the ball hit the batsman, or part of his equipment other than the bat, before hitting the bat?
   D Was the part of the batsman or his equipment which was hit by the ball, between the wickets when it was hit? E Was the part of the batsman or his equipment which was hit by the ball, outside of the wicket on the off side when it was hit? F Was the batsman attempting to play a stroke?
   The law can be interpreted as saying that the batsman is out lbw if \([ ( \mathrm { A } \wedge \mathrm { B } ) \vee ( \mathrm { A } \wedge \mathrm { C } ) ] \wedge [ \mathrm { D } \vee ( \mathrm { E } \wedge \sim \mathrm { F } ) ]\).
   The tournament's umpiring manual, in attempting to simplify the law, states that the batsman is out lbw if \(\mathrm { A } \wedge ( \mathrm { B } \vee \mathrm { C } ) \wedge ( \mathrm { D } \vee \mathrm { E } ) \wedge ( \mathrm { D } \vee \sim \mathrm { F } )\). For an lbw decision this requires 4 conditions each to be true.
   1. Use the rules of Boolean algebra to show that the manual's rule is logically equivalent to the law as stated above, naming the rules used at each step. A trainee umpire, using the manual, considers each condition in turn and judges that the following are true: A; B; E; D.
   2. What is her decision and why?
   3. What is odd about her judgement, and does this make the logic invalid?

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.

1

1. A graph is simple if it contains neither loops nor multiple arcs, ie none of the following: \includegraphics[max width=\textwidth, alt={}, center]{a09472cd-8f65-4cca-9683-c386053e66aa-2_81_134_219_1683}
   or \includegraphics[max width=\textwidth, alt={}, center]{a09472cd-8f65-4cca-9683-c386053e66aa-2_79_589_301_328} In an examination question, students were asked to describe in words when a graph is simple. Mark the following responses as right or wrong, giving reasons for your decisions if you mark them wrong.
   1. A graph is simple if there are no loops and if two nodes are connected by a single arc.
   2. A graph is simple if there are no loops and no two nodes are connected by more than one arc.
   3. A graph is simple if there are no loops and two arcs do not have the same ends.
   4. A graph is simple if there are no loops and there is at most one route from one node to another.
2. The following picture represents a two-way switch \includegraphics[max width=\textwidth, alt={}, center]{a09472cd-8f65-4cca-9683-c386053e66aa-2_149_138_932_1119} It can either be in the up state \includegraphics[max width=\textwidth, alt={}, center]{a09472cd-8f65-4cca-9683-c386053e66aa-2_104_104_1128_790}
   or in the down state • or in the down state . Two switches can be used to construct a circuit in which changing the state of either switch changes the state of a lamp. \includegraphics[max width=\textwidth, alt={}, center]{a09472cd-8f65-4cca-9683-c386053e66aa-2_309_543_1448_762} Georgios tries to connect together three two-way switches so that changing the state of any switch changes the state of the lamp. His circuit is shown below. The switches have been labelled 1,2 and 3. \includegraphics[max width=\textwidth, alt={}, center]{a09472cd-8f65-4cca-9683-c386053e66aa-2_496_547_1946_760}
   1. List the possible combination of switch states and determine whether the lamp is on or off for each of them.
   2. Say whether or not Georgios has achieved his objective, justifying your answer.
3. Use a truth table to show that \(( \mathrm { A } \wedge ( \mathrm { B } \vee \mathrm { C } ) ) \vee \sim ( \sim \mathrm { A } \vee ( \mathrm { B } \wedge \mathrm { C } ) ) \Leftrightarrow \mathrm { A }\).

2

1. Given that \(a\) and \(b\) are real numbers, find a counterexample to disprove the statement that, if \(a > b\), then \(a ^ { 2 } > b ^ { 2 }\).
2. A student writes the statement that \(\sin x ^ { \circ } = 0.5 \Longleftrightarrow x ^ { \circ } = 30 ^ { \circ }\).
   1. Explain why this statement is incorrect.
   2. Write a corrected version of this statement.
3. Prove that the sum of four consecutive multiples of 4 is always a multiple of 8 .

3 In each of the following cases choose one of the statements $$P \Rightarrow Q \quad P \Leftarrow Q \quad P \Leftrightarrow Q$$ to describe the relationship between \(P\) and \(Q\).

1. \(P : y = 3 x ^ { 5 } - 4 x ^ { 2 } + 12 x\) \(Q : \frac { \mathrm { d } y } { \mathrm {~d} x } = 15 x ^ { 4 } - 8 x + 12\)
2. \(\quad P : x ^ { 5 } - 32 = 0\) where \(x\) is real \(Q : x = 2\)
3. \(\quad P : \ln y < 0\) \(Q : y < 1\)

11 Line 8 states that \(\frac { a + b } { 2 } \geqslant \sqrt { a b }\) for \(a\), \(b \geqslant 0\). Explain why the result cannot be extended to apply in each of the following cases.

1. One of the numbers \(a\) and \(b\) is positive and the other is negative.
2. Both numbers \(a\) and \(b\) are negative.

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]