Logic STUDY GUIDE

5. 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)

(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)

17. 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)

1

(Total 8 marks)

6. 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.”

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

(p q) p.

(Total 4 marks)

11. 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.

(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

(Total 4 marks)

21. 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.

2

(Total 6 marks)

16. 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)

18. (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.

(Total 8 marks)

3

24. (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)

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

(Total 4 marks)

26. 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)

29. 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.

4

(Total 6 marks)

Date/page #: _____________

LogicSTUDY GUIDE – hmwk quiz

6. 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.”

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

(p q) p.

(Total 4 marks)

17. 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)

5

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)

(Total 8 marks)

24. (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)

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

(1)(Total 4 marks)

26. 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)

6

29. 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 ii) F iv)

T F F i) T T T

F T T F T T T

F F T T F iii) 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.

(Total 6 marks)

7

LogicReview Part 1 – Answer Key

5. (a) (i) p q (1 mark)

(ii) If Matthew doesn’t cook dinner then Jill will not wash the dishes (1 mark)

(b) (i)

(1 mark for each correct column).

(ii) The truth table is showing that the following argument is valid.If Matthew arrives home before six o’clock he will cook dinner.If Matthew cooks dinner then Jill will wash the dishes. Jill didnot wash the dishes. Therefore Matthew did not arrive beforesix o’clock. (2 marks for correct reasoning)

6. (a) (p ¬q) (1 mark)8

p q q r ¬r (p q) (q r) ¬r ¬p

T T F F F

T F T F F

F T F F F

F T T F F

T T F F T

T F T F T

T T F F T

T T T T T

(b) If you have understood this topic and will not be able to do thisquestion, then you have not understood this topic.

(1 mark for of ¬q, ¬p, and total of 3 marks)

11. (a) (i) ¬(p q) alternatively ¬p ¬q (1 mark)

(ii) ¬(p q) alternatively ¬p ¬q (1 mark)

(b)¬p ¬p qF TF FT TT T

(2 marks – 1 for each column).

16. (a) (i) ¬q ¬p (1 mark)

(ii) q (p r) (1 mark for q and 1 mark (p r) for WITH parentheses)

(b) If Alex does not play the flute then it is not true that he is a scientistor from Uruguay.ORIf Alex does not play the flute then he is neither a scientist norfrom Uruguay. (1 mark for ‘if…then…, 1 mark for antecedent, 1 mark for consequent)

(c)q p ¬(q p) ¬r ¬(q p)

T F TT F FT F TT F FT F TT F FF T TF T T

(1 mark for each correct column – total of 3 marks)

Not a logically valid argument. (1 mark)

9

17. a – T, b – T, c – F, d – T, e – T, f – F, g – T, h – T (1 mark for each correct value)Note: Award (A1) for each correct answer.

18. (a) 2x = 2x = 1 (2 marks)

(b) Yes (1 mark)

(c) x2 = x 2x + 3 = 5

(1 mark for x2 = x, 1 mark for or verbal If…then…, 1 mark for 2x + 3 = 5)

(d) x = 0 (2 marks)

21. (a) Paula eats chocolates and does not watch television(1 mark for “and”, 1 mark for the rest)

(b) Paula eats chocolates or watches television but not both(1 mark for “or”, 1 mark for “but not both)

(c) If Paula watches television then she does not eat chocolates (1 mark for If…then…, 1 mark for antecedent and consequent)

24. (a)¬p (p q) ¬p

q

F F TF F TT T TT F T (1 mark for each

correct column)

(b) Valid argument or tautology (do not accept “true”) (1 mark)

29. (a) Both are “p or q”, the first is “but not both” (1 mark)

(b)

q p q p q p q p q (1 mark for each value)

F T

T

10

F

(c) Tautology. (1 mark)

11