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IB Math Studies – Chapter 15 – Logic – Review Questions
1.
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.
Working:
Answers:
(a) ...................................................
...................................................
...................................................
(b) ...................................................
...................................................
...................................................
(c) ...................................................
...................................................
...................................................
(Total 6 marks)
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
(ii)
(2)
 r  (p  q)
(2)
(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)
(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)
3.
Two propositions p and q are defined as follows.
1
(a)
(b)
p: Jones passed this course
q: Smith passed this course
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.
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)
4.
Consider the following statements.
p: students work hard
q: 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.
(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)
5.
(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.
(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)
1
p
T
T
F
F
contrapositive
2
3
q
pq
T
T
F
F
T
F
F
T
4
p
F
F
T
F
5
pq
T
T
6
(p  q)  ( p   q)
F
F
F
F
Working:
Answer:
(c) ..................................................................
(Total 8 marks)
6.
Consider the following logic statements:
2
(a)
p: the train arrives on time
q: I am late for school
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."
(c)
Complete the following truth table.
p
q
p
T
T
F
T
F
F
F
T
T
F
F
T
q
F
T
F
T
pq
F
T
–
T
pq
F
–
–
T
(d)
Are the two compound propositions (p   q) and ( p   q) logically equivalent?
Working:
Answers:
(a) ..................................................................
..................................................................
(b) ..................................................................
(d) ..................................................................
(Total 8 marks)
7.
Write down the values for a, b, c, d, e and f from the table below:
p
q
p
pq
pq
T
T
a
T
F
b
F
T
c
F
F
pq
d
pq
pq
f
e
(Total 6 marks)
8.
(a)
Copy and complete the table below by filling in the three empty columns.
p
q
pq
pq
p
(p  q)   p
T
T
T
T
T
F
F
T
F
T
F
T
F
F
F
F
(p  q)   p  q
(3)
(b)
What word is used to describe the argument (p  q)   p  q?
(1)
(Total 4 marks)
9.
You may choose from three courses on a lunchtime menu at a restaurant.
3
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.
(c)
Complete the following truth table.
(2)
s
T
T
F
F
d
T
F
T
F
s
sd
(2)
Working:
Answers:
(a) ...................................................
(b) ...................................................
...................................................
...................................................
(Total 6 marks)
10.
Consider the following logic propositions:
p: Sean is at school
q: Sean is playing a game on his computer.
(a)
Write in words, p  q.
(b)
Write in words, the converse of p  q.
(c)
Complete the following truth table for p  q.
p
q
T
T
T
F
F
T
F
F
(2)
(2)
q
p  q
(2)
Working:
Answers:
(a) ...................................................
...................................................
...................................................
(b) ...................................................
...................................................
...................................................
...................................................
(Total 6 marks)
11.
(a)
(b)
Solve 2x + 3 = 5.
Consider the logic statements.
4
p: 2x + 3 = 5
2
q: x = x
2
The compound proposition 2x + 3 = 5  x = 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)
12.
Consider the statements
(a)
(b)
(c)
p: The sun is shining.
q: I am wearing my hat.
Write down, in words, the meaning of q   p.
Complete the truth table.
p
q
p
T
T
T
F
F
T
F
F
Write down, in symbols, the converse of q   p.
qp
Working:
Answers:
(a) ..................................................
(c) ..................................................
(Total 6 marks)
IB Math Studies – Chapter 15 – Logic – Review Questions Mark Scheme
1.
(a)
(b)
(c)
Paula eats chocolates and does not watch television
For “and”
For the rest correct
(A1)
(A1)
(C2)
Paula eats chocolates or watches television but not both
For correct...or...
For “but not both”
(A1)
(A1)
(C2)
If Paula watches television then she does not eat chocolates
for if....then
for antecedent and consequent both correct
(A1)
(A1)
(C2)
[6]
2.
(a)
(i)
“The food supply is adequate and the visitors are hungry but the
oven is not working,” (or equivalent statement).
(A2)
5
“Either the oven is working and the food supply is adequate
or the visitors are not hungry,” (or equivalent statement).
(ii)
(b)
(p  q)  (p  q)
Notes: Award (A1) for (p  q) and (p  q), (A1) for .
(A2)
4
(A2)
2
(R1)
3
(c)
p
q
(p  q)
(p  q)
(p  q)  (p  q)
T
T
T
T
T
T
F
F
T
T
F
T
F
T
T
F
F
(A2)
F
F
T
Therefore, (p  q)  (p  q) is a tautology
Notes: Follow through from part (ii) (b).
Award [½ mark] for each correct bold column and round up.
Award (R1) for a correct conclusion based on truth values in column headed (p  q)  (p 
q).
[9]
3.
(a)
¬(p  q) alternatively ¬p  ¬q
¬(p  q) alternatively ¬p  ¬q
(i)
(ii)
(A1)
(A1)
(b)
p
T
T
F
F
q
¬p
¬p  q
T
F
T
F
F
F
T
T
T
F
T
T
Note: Award (A1) for each bold column.
(A2)
[4]
4.
(a)
(b)
¬p  ¬q
p
T
T
F
F
q
T
F
T
F
¬p
F
F
T
T
(A3)
(C3)
(A5)
(C5)
¬p  ¬q
T
T
F
T
¬q
F
T
F
T
(A1)
(A1)
(A1)

No it is not valid (A2)
[8]
5.
(a) and (b)
1
2
3
4
p
q
¬p
pq
T
T
T
F
T
F
F
F
F
T
F
T
F
F
(T) (A2)
(F) (A2)
(c)
The last column is a contradiction.
5
pq
T
T (A1)
T
F (A1)
6
(p  q)  (¬p  ¬q)
F
F
F
F
(C6)
(A2)
(C2)
[8]
6.
(a)
If the train arrives on time then I am not late for school.
(A2)
(C2)
6
Note: Award (A1) for “If... then” in correct order and (A1) for “not”.
(b)
¬p  ¬q
(A2)
(C2)
Note: Award (A1) for both ¬ symbols and (A1) for .
(c)
missing elements are
T
F
F
(A3)
(C3)
(d)
The implications are not logically equivalent.
(A1)
(C1)
[8]
7.
a=F
b=F
c=T
d=F
e=T
f=F
(A1)
(A1)
(A1)
(A1)
(A1)
(A1)
[6]
8.
(a)
p
T
T
F
F
(b)
q
T
F
T
F
¬p
q
pq
pq
(p  q)  ¬p

T
T
T
F
F
T
F
T
F
F
F
T
F
T
T
T
T
T
F
F
F
(A1)(A1)(A1)
3
T
F
T
Note: A correct column always receives (A1). If a column is incorrect, ft at each subsequent
step.
Whole table does not need to be redrawn, just the 3 columns is enough.
Valid argument or tautology (do not accept “true”)
Note: Follow through from candidate’s table as follows: If the implication column is empty
award A1 for tautology otherwise ft must be appropriate: invalid or not valid for mixed
column or contradiction for four Fs.
(A1)
1
[4]
9.
(a)
(b)
m  (s  d)
Notes: (A1) for m 
(A1) for (s  d)
(A1)(A0) if brackets are missing
OR
(m  s)  (m  d)
Notes: (A1) for both brackets correct, (A1) for disjunctive “or” (A1)(A0) if brackets are missing.
(A2)
If you choose a salad then you do not choose a dessert.
Notes: (A1) for “if …then…”
(A1) for salad and no dessert in the correct order.
OR
If you choose a salad you do not choose a dessert.
(A2)
(A2)
(A2)
(C2)
(C2)
(c)
s
T
T
F
F
d
s
s  d
T
F
T
F
F
T
T
T
T
F
T
F
Note: (A1) for each correct column
(A1)(A1)(ft)
(C2)
[6]
10.
(a)
Either Sean is at school or Sean is playing a game on his
computer but not both.
(A1)(A1)
7
Note: (A1) for ‘either ... or but not both’ (A1) for correct
statements. ‘Either’ can be omitted.
(b)
If Sean is not playing a game on his computer then Sean is at school.
Note: (A1) for ‘If ... then’ (A1) for correct propositions
in the correct order.
(C2)
(A1)(A1)
(C2)
(c)
q
F
T
F
T
p q
F
T
T
T
Note: (A1) for each correct column.
(A1)(A1)(ft)
(C2)
[6]
11.
(a)
2x = 2
x=1
(M1)
(A1)
(C2)
(b)
Yes
(A1)
(C1)
(c)
2
x = x  2x + 3 = 5
(A1)(A1)(A
(C3)
1)
2
Note: Award (A1) for x = x, (A1) for , (A1) for 2x + 3 = 5.
2
Accept also: If x = x (A1) then (A1) 2x + 3 = 5 (A1).
(d)
x=0
(A2)
(C2)
(a)
If I am wearing my hat (then) the sun is not shining.
[8]
12.
(A1)(A1) (C2)
Notes: (A1) for if with or without then
(A1) for both correct statements with implication in correct direction. Use of ‘therefore’ is
incorrect.
(b)
p
T
T
F
F
(c)
q
T
F
T
F
p
F
F
T
T
(A1)
 p  q (or q   p)
Notes: Do not split the (A2)
Accept only these two statements.
qp
F
T
T
T
(A1)(ft)
(C2)
(A2)
(C2)
[6]
8