Converse, Inverse, Contrapositive Revision 1

1a. [1 mark]

Markscheme

I am tired and I am happy       (A1) (C1)

Note: Accept “I am happy and I am tired”.

[1 mark]

1b. [1 mark]

Markscheme

¬ ¬⇒        (A1) (C1)

Note: Accept alternate forms of logic notation.

[1 mark]

1c. [2 marks]

Markscheme

¬ ⇒        (A1)(A1) (C2)

Note: Award (A1) for seen between their antecedent and consequent, and ⇒ (A1) for completely

correct expression.

[2 marks]

1d. [2 marks]

Markscheme

1

  (A1)(A1) (C2)

Note: Award (A1) for each correct column.

[2 marks]

2a. [3 marks]

Markscheme

    (A1)(A1)(A1)

 

Note:     Award (A1) for “ ”.

Award (A1) for “ ” or “ ” (or “ ”or “ ”)

Award (A1) for “ ”.

Award at most (A1)(A1)(A0) if parentheses are missing for .

Award (A0)(A0)(A1) for .

 

[3 marks]

2b. [3 marks]

Markscheme

is not a multiple of 5 and ( ) is (either) a factor of 60 or ( ) is a multiple of 4, but not both     (A1)(A1)

(A1)

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Note:     Award (A1) for “ is not a multiple of 5”, (A1) for “( ) is a factor of 60 or ( ) is a multiple of 4

but not both”, (A1) for “and” in the correct position. Accept only “but not both” in the second (A1).

Award at most (A1)(A1)(A0) for using extra statements such as “If ...then”, “if and only if” etc.

 

[3 marks]

2c. [3 marks]

Markscheme

     (A1)(A1)(A1)(ft)

 

Note:     Award (A1) for each correct column. Last column follows through from previous two.

 

[3 marks]

2d. [1 mark]

Markscheme

because not all the entries in the column are F     (R1)(ft)

 

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Note:     If all entries in the last column of their truth table are T, award (R1)(ft) for an answer of “it is a

tautology”. Only award (R1)(ft) if the column is identified in the justification.

 

[1 mark]

2e. [1 mark]

Markscheme

accept one of: 1 OR 2 OR 3 OR 6     (A1)

 

Note:     Award (A1) for any one of the above answers.

 

[1 mark]

3a. [2 marks]

Markscheme

i)      is not a rational number        (A1)

Note: Accept “ is an irrational number”.

 

ii)   any non-rational number (for example: , …)       (A1) (C2)

3b. [3 marks]

Markscheme

      (A1)(A1)(A1) (C3)

Note: Award (A1) for “ ” seen, (A1) for “ ” as the consequent and (A1) for “ ” or “ ”

as the antecedent (the parentheses are required).

3c. [1 mark]

Markscheme

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all integers are rational numbers (and therefore  cannot be an integer if it is not a rational number)    

(R1)

Note: Accept equivalent expressions.

OR

if is an integer, then  is a rational number, therefore if  is not a rational number, then  is not an

integer (contrapositive)   (R1) (C1)

Note: Accept “If  is not in , then  is not in ” with a Venn diagram showing ,  and  correctly.

4a. [1 mark]

Markscheme

is not a multiple of      (A1)     (C1)

4b. [2 marks]

Markscheme

    (A1)(A1)(C2)

Note: Award (A1) for , (A1) for and in the correct order.

Accept .

4c. [1 mark]

Markscheme

Converse     (A1) (C1)

4d. [2 marks]

Markscheme

not valid     (A1)

for example  is a multiple of  and not a multiple of      (R1)     (C2)

 

Notes: Do not award (A1)(R0). Any multiple of 6 that is not a multiple of  can be accepted as a

counterexample.

5

5a. [2 marks]

Markscheme

    (A1)(A1)     (C2)

 

Note: Award the first (A1) for seeing the implication sign, the second (A1) is for a correct answer only.

Not using the implication earns no marks.

 

[2 marks]

5b. [1 mark]

Markscheme

    (A1)(ft)     (C1)

 

Note: Award (A1)(ft) where the propositions in the implication in part (a) are exchanged.

 

[1 mark]

5c. [2 marks]

Markscheme

Not equivalent; a kite or an isosceles trapezium (for example) can have diagonals that are equal in

length.     (A1)(R1)     (C2)

 

Notes: Accept a valid sketch as reasoning.

     If the reason given is that a square has diagonals of equal length, but is not a rectangle, then award

(R1)(A0).

     Do not award (A1)(R0).

     Do not accept solutions based on truth tables.

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[2 marks]

5d. [1 mark]

Markscheme

Inverse     (A1)     (C1)

 

Note: Do not accept symbolic notation.

 

[1 mark]

6a. [2 marks]

Markscheme

If Eva is losing weight then Eva is on a diet     (A1)(A1)     (C2)

 

Notes: Award (A1) for If… then…

     For Spanish candidates, only accept “Si” and “entonces”.

     For French candidates, only accept “Si” and “alors”.

     For all 3 languages these words are from the subject guide.

     Award (A1) for correct propositions in correct order.

 

[2 marks]

6b. [2 marks]

Markscheme

If Eva is not on a diet then she is not losing weight     (A1)(A1)     (C2)

 

7

Notes: Award (A1) for “not on a diet” and “not losing weight” seen, (A1) for complete correct answer.

     No follow through from part (a).

 

[2 marks]

6c. [2 marks]

Markscheme

The statements are logically equivalent     (A1)(ft)

The contrapositive is always logically equivalent to the original statement     (R1)(ft)

OR

A correct truth table showing the equivalence     (R1)(ft)     (C2)

 

Note: Follow through from their answers to part (a) and part (b).

 

[2 marks]

7a. [2 marks]

Markscheme

If I do not have a bowl of soup then I have an ice cream.     (A1)(A1)     (C2)

 

Notes: Award (A1) for If… then…

     Award (A1) for correct statements in correct order.

 

[2 marks]

7b. [2 marks]

Markscheme

8

     (A1)(A1)(ft)     (C2)

 

Note: Follow through from third column to fourth column.

 

[2 marks]

7c. [2 marks]

Markscheme

    (A1)(A1)     (C2)

 

Notes: Award (A1) for .

     Award (A1) for and in correct order.

     Accept .

 

[2 marks]

8a. [3 marks]

Markscheme

(i)      (A1)(A1)

Note: Award (A1) for conjunction, (A1) for negation of q.

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(ii) OR      (A1)     (C3)

8b. [3 marks]

Markscheme

If Yuiko is not studying Chinese, (then) she is studying French.     (A1)(A1)(A1)     (C3)

Notes: Award (A1) for “if … (then)” seen, award (A1) for “not studying Chinese” seen, (A1) for correct

propositions in correct order.

9a. [2 marks]

Markscheme

Carlos is not playing the guitar and he is studying for his IB exams.     (A1)(A1)     (C2)

Note: Award (A1) for “and”, (A1) for correct statements.

[2 marks]

9b. [1 mark]

Markscheme

     (A1)     (C1)

[1 mark]

9c. [3 marks]

Markscheme

    (A1)(A1)(A1)     (C3)

Notes: Award (A1) for implication, (A1) for the , (A1) for both and in the correct order. If

correct converse seen in words only award (A1)(A1)(A0). Accept . Accept for .

[3 marks]

10a. [2 marks]

10

Markscheme

If (both) the numbers x and y are even (then) the sum of x and y is an even number.     (A1)(A1)     (C2)

Note: Award (A1) for If…(then), (A1) for the correct statements in the correct order.

[2 marks]

10b. [2 marks]

Markscheme

If (both) the numbers x and y are not even (then) the sum of x and y is not an even number.     (A1)

(A1)     (C2)

Notes: Award (A1) for If…(then), (A1) for the correct not p, and not q in the correct order. Accept the

word odd for the phrase “not even”.

[2 marks]

10c. [2 marks]

Markscheme

The inverse of a statement is not (necessarily) true, because two odd (not even) numbers, always have

an even sum.     (A1)(R1)(ft)    (C2)

Notes: Award (A1)(R1) if a specific counter example given instead of a reason stated in general terms,

e.g. the inverse is not true because, 5 and 7 have an even sum. Do not award (A1)(R0). Follow through

from their statement in part (b).

[2 marks]

11a. [2 marks]

Markscheme

11

     (A1)(A1)(ft)     (C2)

Notes: Award (A1) for each correct column. Follow through in 4th column from their 3rd column.

[2 marks]

11b. [2 marks]

Markscheme

    (A1)(A1)     (C2)

Note: Award (A1) for and p in any order, (A1) for .

[2 marks]

11c. [2 marks]

Markscheme

If I can swim 50 metres (then) I do not take swimming lessons.     (A1)(A1)     (C2)

Note: Award (A1) for If… (then), (A1) for correct propositions in the correct order.

[2 marks]

12a. [4 marks]

Markscheme

12

     (A1)(A1)(A1)(ft)(A1)(ft)    

(C4)

Notes: Award (A1) for each correct column.

            Award first (A1)(ft) from their third column in the table.

            Award second (A1)(ft) from their fourth and fifth column in the table.

[4 marks]

12b. [1 mark]

Markscheme

Tautology     (A1)(ft)     (C1)

Note: Answer must be consistent with last column in table.

[1 mark]

12c. [1 mark]

Markscheme

All entries (in the final column) are true.     (R1)(ft)     (C1) 

Note: Answer must be consistent with their answer to part (b)(i).

Note: Special case (A1)(R0) may be awarded.

[1 mark]

13a. [3 marks]

Markscheme

    (A1)(A1)(A1)     (C3)

13

Note: Award (A1) for two conjunctions, (A1) for negation seen on , (A1) for correct compound

statement.

[3 marks]

13b. [3 marks]

Markscheme

If I visited (either) Sarah’s Snackbar or Pete’s Eats (then) I did not visit Alan’s Diner.     (A1)(A1)(A1)    

(C3) 

Note: Award (A1) for If (then), (A1) for Sarah’s Snackbar or Pete’s Eats, (A1) for did not visit Alan’s

Diner.

[3 marks]

14a. [3 marks]

Markscheme

If I do not choose history then I choose either psychology or I choose art     (A1)(A1)(A1)     (C3)

Notes: Award (A1) for ‘if… (then)…’

Award (A1) for ‘not choose history.’

Award (A1) for ‘choose (either) psychology or art (or both).’

If the order of the statements is wrong award at most (A1)(A1)(A0).

[3 marks]

14b. [1 mark]

Markscheme

14

     (A1)     (C1)

[1 mark]

14c. [2 marks]

Markscheme

Neither, because not all the entries in the last column are the same.     (A1)(ft)(R1)     (C2)

Notes: Do not award (R0)(A1). Follow through from their answer to part (b). Reasoning must be

consistent with their answer to part (b).

[2 marks]

15a. [2 marks]

Markscheme

If a quadrilateral is not a square (then) the four sides of the quadrilateral are not equal.     (A1)(A1)    

(C2)

Note: Award (A1) for “if…(then)”, (A1) for the correct phrases in the correct order.

 

[2 marks]

15b. [2 marks]

Markscheme

If the four sides of the quadrilateral are equal (then) the quadrilateral is a square.     (A1)(A1)(ft)     (C2)

15

Note: Award (A1) for “if…(then)”, (A1)(ft) for the correct phrases in the correct order.

 

Note: Follow through in (b) if the inverse and converse in (a) and (b) are correct and reversed.

 

[2 marks]

15c. [2 marks]

Markscheme

The converse is not always true, for example a rhombus (diamond) is a quadrilateral with four equal

sides, but it is not a square.     (A1)(R1)     (C2)

Note: Do not award (A1)(R0).

 

[2 marks]

16a. [2 marks]

Markscheme

If the sun is shining then I will go swimming.     (A1)(A1)     (C2)

 

Note: Award (A1) for “if…then” and (A1) for correct order.

 

[2 marks]

16b. [2 marks]

Markscheme

Either the sun is not shining or I will go swimming.     (A1)(A1)     (C2)

16

Note: Award (A1) for both correct statements and (A1) for “either” “…or”.

 

[2 marks]

16c. [1 mark]

Markscheme

     (A1)     (C1)

[1 mark]

16d. [1 mark]

Markscheme

They are (logically) equivalent.     (A1)     (C1)

Note: Do not accept any other answers.

 

[1 mark]

17a. [2 marks]

Markscheme

     (A1)(A1)(ft)     (C2)

Note: (A1) for each correct column.

17

[2 marks]

17b. [2 marks]

Markscheme

It is not true that food or drinks may be taken into the cinema.

Note: (A1) for “it is not true”. (A1) for “food or drinks”.

OR

Neither food nor drinks may be taken into the cinema.

Note: (A1) for “neither”. (A1) for “nor”.

OR

No food and no drinks may be taken into the cinema.

Note: (A1) for “no food”, “no drinks”. (A1) for “and”.

OR

No food or drink may be brought into the cinema.     (A2)     (C2)

Note: (A1) for “no”, (A1) for “food or drink”. Do not penalize for use of plural/singular.

Note: the following answers are incorrect:

No food and drink may be brought into the cinema. Award (A1) (A0)

Food and drink may not be brought into the cinema. Award (A1) (A0)

No food or no drink may be brought into the cinema. Award (A1) (A0)

[2 marks]

17c. [2 marks]

Markscheme

Note: (A1) for both negations, (A1) for conjunction.

18

OR

     (A1)(A1)     (C2)

Note: (A1) for negation, (A1) for in parentheses.

[2 marks]

18a. [2 marks]

Markscheme

Either Sean is at school or Sean is playing a game on his computer but not both.     (A1)(A1)     (C2)

Note: (A1) for ‘either ... or but not both’ (A1) for correct statements. ‘Either’ can be omitted.

[2 marks]

18b. [2 marks]

Markscheme

If Sean is not playing a game on his computer then Sean is at school.     (A1)(A1)     (C2)

Note: (A1) for ‘If ... then’ (A1) for correct propositions in the correct order.

[2 marks]

18c. [2 marks]

Markscheme

     (A1)(A1)(ft)     (C2)

Note: (A1) for each correct column.

19

[2 marks]

Printed for Carlucci American International School of Lisbon

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