1. Three propositions p, q and r are defined as follows:

p: the water is cold. q: the water is boiling. r: the water is warm.

(a) Write one sentence, in words, for the following logic statement:

( p q) r

(b) Write the following sentence as a logic statement using symbols only.

"The water is cold if and only if it is neither boiling nor warm"

Working:

Answers:

(a) …………………………………………..(b) …………………………………………..

(Total 4 marks)

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2. Three propositions are defined as follows:

p: The oven is working.

q: The food supply is adequate.

r: The visitors are hungry.

(a) Write one sentence, in words only, for each of the following logic statements.

(i) q r p(2)

(ii) r (p q)(2)

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(b) Write the sentence below using only the symbols p, q and logic connectives.

"If the oven is working and the food supply is adequate then the oven is working or the food supply is adequate."

(2)

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(c) A tautology is a compound statement which is always true. Use a truth table to determine whether or not your answer to part (b) is a tautology.

Hint: Begin by writing the first two columns of your truth table in the following format:

p q

T T

T F

F T

F F(3)

(Total 9 marks)

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3. Consider the following statements:

p: Good mathematics students go to good universities

q: Good music students are good mathematics students

r: Students who go to good universities get good jobs

(a) From these statements, write two valid conclusions.

(b) Write in words each of the following

(i) q;

(ii) p r.

Working:

Answers:(a) …………………………………………..

…………………………………………..…………………………………………..

(b) (i) ……………………………………...……………………………………...

(ii) ……………………………………..……………………………………..

(Total 4 marks)

4. Let the propositions p, q and r be defined as:

p: Matthew arrives home before six o’clockq: Matthew cooks dinnerr: Jill washes the dishes

(a) (i) Express the following statement in logical form.

If Matthew arrives home before six o’clock then he will cook dinner.(1)

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(ii) Write the following logic statement in words.

q r(1)

(b) (i) Copy and complete the truth table below.

p q r p q q r r (p q) (q r) r p [(p q) (q r) r] p

T T T T

T T F T

T F T T

T F F T

F T T T

F T F T

F F T T

F F F T(5)

(ii) Explain the significance of the truth table above.(2)

(Total 9 marks)

5. The propositions p and q are defined as follows:

p: you have understood this topic

q: you will be able to do this question

(a) Write the following proposition in symbols using p, q and logical connectives only.

“You have understood this topic, or you will not be able to do this question.”

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(b) Explain, in words only, what the following symbolic proposition represents:

(p q) p.

Working:

Answers:

(a) …………………………………………..(b) …………………………………………..

(Total 4 marks)

6. The propositions p, q and r are defined as follows:

p: this is a good courseq: the course is worth takingr: the grading is lenient

(a) Write a symbolic statement for each of the following sentences.

(i) If this is a good course, then it is worth taking.

(ii) Either the grading is lenient, or the course is not worth taking.(2)

(b) Write the following argument using p, q, r and logic symbols or connectives only.

If this is a good course, then it is worth taking. Either the grading is lenient, or the course is not worth taking. But the grading is not lenient. Therefore, this is not a good course.

(2)(Total 4 marks)

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7. If each of the following compound propositions is true, what conclusions can be made?

(a) x < 7 or x 3, and x < 7

(b) p = 3 if and only if q = 5, and if q 5 then r 12.

Working:

Answers:

(a) …………………………………………..(b) ..................................................................

(Total 4 marks)

8. [(p q) p] q

(a) Complete the truth table below for the compound statement above.

p q p q (p q) p [(p q) p] q

T T

T F

F T

F F

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(b) Explain the significance of your result.

Working:

Answers:

(b) ....................................................................................................................................

(Total 4 marks)

9. Two propositions p and q are defined as follows:

p: the number ends in zero

q: the number is divisible by 5

(a) Write in words

(i) p q;

(ii) the converse of (p q).

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(b) Write in symbolic form

(i) the inverse of (p q);

(ii) the contrapositive of (p q).

Working:

Answers:

(a) (i) ......................................................................................................................

(ii) ......................................................................................................................

(b) (i) ...........................................................(ii) ...........................................................

(Total 4 marks)

10. Two propositions p and q are defined as follows.

p: Jones passed this courseq: Smith passed this course

(a) Write in symbolic form

(i) neither Jones nor Smith passed the course;

(ii) it is not the case that Jones and Smith both passed the course.

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(b) Complete the following truth table for the logic statement p q.

p q p p q

T T

T F

F T

F F

Working:

Answers:

(a) (i) ……………………………………...(ii) ……………………………………...

(Total 4 marks)

11. Let p and q be the statements

p: you watch the music TV channelq: you like music

(a) Consider the following logic statement.

If you watch the music TV channel then you like music.

(i) Write down in words the inverse of the statement.

(ii) Write down in words the converse of the statement.(4)

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(b) Construct truth tables for the following statements:

(i) p q.

(ii) p q.

(iii) p q.

(iv) p q.(4)

(c) Which of the statements in part (b) are logically equivalent?(1)

(Total 9 marks)

12. Consider the statement “If a figure is a square, then it is a rhombus”.

(a) For this statement, write in words

(i) its converse;

(ii) its inverse;

(iii) its contrapositive.

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(b) Only one of the statements in part(a) is true. Which one is it?

Working:

Answers:

(a) (i) ..............................................................................

(ii) ..............................................................................

(iii) ..............................................................................

(b) ..............................................(Total 8 marks)

13. Consider the following statements.

p: students work hardq: students will succeed

(a) Write the following proposition in symbols using p, q and logical connectives only.

If students do not work hard, then they will not succeed.

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(b) Complete the following truth table, relating to the statement made in part (a), and decide whether the statement is logically valid.

p q

T T

T F

F T

F F

Working:

Answers:

(a) ..................................................................(b) ……………………………………..........

(Total 8 marks)

14. (a) The following truth table contains two entries which are incorrect, one in column three and one in column four. Circle the two incorrect entries.

(b) Fill in the two missing values in column five.

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(c) Which one of the following words could you use to describe the statement represented by the values in the last column (number 6)?

(i) converse

(ii) tautology

(iii) inverse

(iv) contradiction

(v) contrapositive

1 2 3 4 5 6

p q p q p p q (p q) ( p q)

T T T F T F

T F F F F

F T F T T F

F F T F F

Working:

Answer:(c) ..................................................................

(Total 8 marks)

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15. Consider each of the following statements:

p: Alex is from Uruguayq: Alex is a scientistr: Alex plays the flute

(a) Write each of the following arguments in symbols:

(i) If Alex is not a scientist then he is not from Uruguay.

(ii) If Alex is a scientist, then he is either from Uruguay or plays the flute.(3)

(b) Write the following argument in words:

r (q p)(3)

(c) Construct a truth table for the argument in part (b) using the values below for p, q, r and r. Test whether or not the argument is logically valid.

p q r r

T T T F

T T F T

T F T F

T F F T

F T T F

F T F T

F F T F

F F F T(4)

(Total 10 marks)

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16. Consider two propositions p and q. Complete the truth table below for the compound proposition.

(p q) ( p q)

p q p q p q p q (p q) ( p q)

T T F F F (d) T

T F F T (b) F (f)

F T (a) F (c) T (g)

F F T T F (e) (h)

Working:

(Total 8 marks)

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17. (a) Solve 2x + 3 = 5.

(b) Consider the logic statements.

p: 2x + 3 = 5 q: x2 = x

The compound proposition 2x + 3 = 5 x2 = x is given.Is this compound proposition true?

(c) Write down the converse of this compound proposition.

(d) Give an example to show that the converse is false.

Working:

Answers:

(a) ..................................................................(b) ..................................................................(c) ..................................................................

..................................................................(d) ..................................................................

..................................................................

(Total 8 marks)

18. Let p and q be the statements:

p: Sarah eats lots of carrots.q: Sarah can see well in the dark.

Write the following statements in words.

(a) p q.

(b) p q.

(c) Write the following statement in symbolic form.

If Sarah cannot see well in the dark, then she does not eat lots of carrots.

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(d) Is the statement in part (c) the inverse, the converse or the contrapositive of the statement in part (a)?

Working:

Answers:

(a) ....................................................................................................................................

(b) ....................................................................................................................................

(c) ..................................................................(d) ..................................................................

(Total 8 marks)

19. Consider the following logic statements:

p: the train arrives on timeq: I am late for school

(a) Write the expression p q as a logic statement.

(b) Write the following statement in logic symbols:

"The train does not arrive on time and I am not late for school."

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(c) Complete the following truth table.

p q p q p q p q

T T F F F F

T F F T T –

F T T F – –

F F T T T T

(d) Are the two compound propositions (p q) and ( p q) logically equivalent?

Working:

Answers:

(a) ....................................................................................................................................

(b) ..................................................................(d) ..................................................................

(Total 8 marks)

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20. Two logic propositions are given.

p: Paula eats chocolates.q: Paula watches television.

Write in words

(a) p q;

(b) p q;

(c) q p.

Workin g :

A n sw ers:(a ) ....................... ..... .. .. .. .. .. .. .. ... .. .. ..

...... .. .. .. ....... .. .. .. .. .. .. .. .. .. .. ..... .. .. .. .

...... .. .. .. ....... .. .. .. .. .. .. .. .. .. .. ..... .. .. .. .

(b ) ..... .. .. .. .. .. .. .. .. ... .......... .............. .. .

...... .. .. .. ....... .. .. .. .. .. .. .. .. .. .. ..... .. .. .. .

...... .. .. .. ....... .. .. .. .. .. .. .. .. .. .. ..... .. .. .. .

(c ) ....................... ..... .. .. .. .. .. .. .. ... .. .. ..

...... .. .. .. ....... .. .. .. .. .. .. .. .. .. .. ..... .. .. .. .

...... .. .. .. ....... .. .. .. .. .. .. .. .. .. .. ..... .. .. .. .

(Total 6 marks)

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21. Write down the values for a, b, c, d, e and f from the table below:

p q p p q p q p q p q p q

T T a d

T F b f

F T c

F F e

(Total 6 marks)

22. Let p stand for the proposition “I will walk to school”. Let q stand for the proposition “the sun is shining”.

(a) Write the following statements in symbolic logic form

(i) “If the sun is shining then I will walk to school.”

(ii) “If I do not walk to school then the sun is not shining.”(4)

(b) Write down, in words, the converse of the statement

“If the sun is shining then I will walk to school.”(2)

(Total 6 marks)

23. (a) Copy and complete the table below by filling in the three empty columns.

p q p q p q p (p q) p (p q) p q

T T T T

T F F T

F T F T

F F F F(3)

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(b) What word is used to describe the argument (p q) p q?(1)

(Total 4 marks)

24. Complete the Truth Table for the compound proposition (p q) (p q).

p q q (p q) (p q) (p q) (p q)

T T F F

T F T T

F T F T

F F F F

Workin g :

(Total 8 marks)

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25. Consider the following logic statements:

p: x is a factor of 6

q: x is a factor of 24

(a) Write p q in words.(1)

(b) Write the converse of p q.(1)

(c) State if the converse is true or false and give an example to justify your answer.(2)

(Total 4 marks)

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26. Consider the statements

p: The sun is shining.q: I am wearing my hat.

(a) Write down, in words, the meaning of q p.

(b) Complete the truth table.

p q p q p

T T

T F

F T

F F

(c) Write down, in symbols, the converse of q p.

Wo rk in g :

A n sw ers:

(a ) .... .. .. .. ........... ................ .. .. .. .......

(c ) ........... .. .......... .. ... .. .. .. .. .. .. .. ..... .. .(Total 6 marks)

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27. Two logic propositions are given

p: Dany goes to the cinemaq: Dany studies for the test

(a) Write in words the proposition

p q.

(b) Given the statement s: “If Dany goes to the cinema then Dany doesn’t study for the test”.

(i) Write s in symbolic form.

(ii) Write in symbolic form the contrapositive of part (b)(i).

Wo rk in g :

A nsw ers:

(a ) .... .. .. .... .. .. .. .. .. .. ................ .. .. .. .. .. .

(b ) (i) ............ .. .. .. .. .. ........... .. .. .. .. .. .

(ii) ..... .. .. .. .. .. .. .. .. .. ..... .. .. .. .. ... . .. .(Total 6 marks)

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28. The truth table below shows the truth-values for the proposition

p q p q

p q p q p q p q p q p q

T T F F F

T F F T T T

F T T F T T T

F F T T F T

(a) Explain the distinction between the compound propositions, p q and p q.

(b) Fill in the four missing truth-values on the table.

(c) State whether the proposition p q p q is a tautology, a contradiction or neither.

Wo rking :

A n sw ers:

(a ) ............. .......... .. ... .. .. .. .. .. .. .. .. ... .. ..

.... .. .. .. .. ..... .. .. .. .. .. .. .. .. .. ..... .. .. .......

...... .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. ......... .. .. .

(c ) ............. .......... .. ... .. .. .. .. .. .. .. .. ... .. ..(Total 6 marks)

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29. You may choose from three courses on a lunchtime menu at a restaurant.

s: you choose a salad,m: you choose a meat dish (main course),d: you choose a dessert.

You choose a two course meal which must include a main course and either a salad or a dessert, but not both.

(a) Write the sentence above using logic symbols.(2)

(b) Write in words s d.(2)

(c) Complete the following truth table.

(2)s d s s d

T T

T F

F T

F F

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Wo rking :

A n sw ers:

(a ) ............. .......... .. ... .. .. .. .. .. .. .. .. ... .. ..

(b ) ..... .. .. .. .. .. .. .. ......... .. .. ................ .. .

...... .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. ......... .. .. .

...... .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. ......... .. .. .(Total 6 marks)

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30. Consider the following logic propositions:

p: Sean is at schoolq: Sean is playing a game on his computer.

(a) Write in words, p q.(2)

(b) Write in words, the converse of p q.(2)

(c) Complete the following truth table for p q.

p q q p q

T T

T F

F T

F F(2)

Workin g :

A n sw ers:(a ) ....................... ..... .. .. .. .. .. .. .. ... .. .. ..

.. .. .. .. .. .. ... .. .. .. .. .. ................ ..........

.. .. .. .. .. .. ... .. .. .. .. .. ................ ..........

(b ) ..... .. .. .. .. .. .. ....... ...................... .. .. .

..... .. .. .. ....... .. .. .. .. .. .. .. .. .. ....... .. .. .. ..

...... .. .. .. ....... .. .. .. .. .. .. .. .. .. .. ..... .. .. .. .

...... .. .. .. ....... .. .. .. .. .. .. .. .. .. .. ..... .. .. .. .(Total 6 marks)

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31. (a) (i) Complete the truth table below.

p q p q (p q) p q p q

T T F F

T F F T

F T T F

F F T T

(ii) State whether the compound propositions (p q) and p q are equivalent.(4)

Consider the following propositions.

p: Amy eats sweetsq: Amy goes swimming.

(b) Write, in symbolic form, the following proposition.

Amy either eats sweets or goes swimming, but not both.(2)

Wo rkin g :

A n sw ers:

(a ) (i i) .... .. .. .. .. .. .. ..... .. .. .. .. ...............

(b ) ..... .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. ............(Total 6 marks)

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32. B and C are subsets of a universal set U such that

U = {x : x , 0 x 10}, B = {prime numbers 10}, C = {x : x , 1 x 6}.

(a) List the members of sets

(i) B

(ii) C B

(iii) B C′

Consider the propositions:

p: x is a prime number less than 10.q: x is a positive integer between 1 and 7.

(b) Write down, in words, the contrapositive of the statement, “If x is a prime number less than 10, then x is a positive integer between 1 and 7.”

Wo rkin g:

A nsw ers:

(a )

(b ) .... ................ .. .. .. ............... .. .. .. .. ..

(i) .... ................ .. . .. ................ .

(i i) ........... .. .. .. .. .. .. .. .. .. .. ... .. .. .. .

(i ii) .... .. .. .. .. .. .. .. ......... .. .. .. .. .. .. .

.. .. .. .. .. .. ............. .. .. .. .. ............... .. .

.. .. .. .. .. .. ............. .. .. .. .. ............... .. .

.. .. .. .. .. .. ............. .. .. .. .. ............... .. .(Total 6 marks)

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