Question Sheet 5
1) Let U = cfw_1, 2, 3, 4, 5, 6. In each of the following cases give examples of
sets A, B, . U such that the equality does not hold.
(i) (A B) C c = A (B C c ),
(ii) A B C = A B (C B),
(iii) (A B) Ac = B,
(iv) (A B)c C = (Ac C) (B c C).
Symbolising Quantied Arguments
1. (i) Symbolise the following argument, given the universe of discourse
is U = set of all animals.
Animals are either male or female.
Not all Cats are male,
Therefore, some cats are female.
Let Cx = x is a cat, M x = x is m
Question Sheet 4
1) Let A = cfw_1, 2, 3, 4. Which of the following statements are true?
(i)
cfw_1 A,
(ii)
6 A,
(iii) cfw_2 A,
(iv) cfw_1, 4, 4 A,
(v)
4 A,
(vi) A A,
(vii) A A,
(viii) cfw_4, 3, 2 A,
(ix) A,
(x) A.
If a statement is false, give a reason.
2)
1 Propositional Logic
Propositions
1.1 Denition
A declarative sentence is a sentence that declares a fact or facts.
Example 1
The earth is spherical.
7 + 1 = 6 + 2
x2 > 0 for all real numbers x.
1 = 0
This sentence is false
These are all declarative sente
Solutions to Additional Questions 6 and 7
6 Arguments in words.
Symbolise the following arguments and prove them valid using the rules
of deduction
1. Adam was the rst man and Eve the rst woman. If the bible is wrong
then Adam wasnt the rst man. If the bi
Question Sheet 3
1) Using truth tables determine whether the following arguments are valid
or invalid.
(i)
p q, q
p
(ii)
p q, q
p,
(iii) p q, p
q,
(iv) p q, q r
p r,
(v) p q, q r
p r,
(vi) p q, q r
(q r) p,
(vii) p q, p r, q r
r.
2) Consider the argument
Solutions to Additional Questions 4 and 5
4 Exercises in C.P.
1. q r, p, p q
r,
1
2
3
4
5
2. q r, p q
p
pq
q
qr
r
A
A
MPP 1,2
A
MPP 3,4
p
pq
q
qr
r
pr
A(CP)
A
MPP 1,2
A
MPP 3,4
CP 1-5
p r,
1
2 |
3 |
4 |
5
6
Note that lines 1-5 are simply the proof for the
Question Sheet 2
1) Let r, s and t denote the propositions
r: Judith goes out for a walk,
s: The moon is out,
t: It is snowing.
Write the following in symbolic form
(i) If the moon is out and it is not snowing then Judith goes out for a
walk,
(ii) It is n
Solutions to Additional Questions 2 and 3
2 Arguments with only one premise.
These can be a little tricky to prove. You often need to use a rule that
increases the numbers of premises, i.e. CP or RAA.
1. A A
A.
1
2 |
3 |
4 |
5
6
2. p
A(RAA)
DN 1
A
MPP 2,3
Question Sheet 1
1) For each of the following decide whether it is a proposition or not, and if
it is, indicate whether it is true or not.
(i)
15 is a positive number,
(ii)
The Earth is at,
(iii)
x2 0,
(iv)
x2 0 for every real number x,
(v)
Shakespeare wr
2 Natural Deduction
A deductive proof is a step-by-step demonstration that a given argument
is valid. At each step we apply rules of inference. We will justify the introduction of these rules by using truth tables to show that they arise from valid
argume
1.3 More Connectives
1.3.1 Conditional and Biconditional
Example 12
Consider the proposition: If it is raining then it is cloudy, which we
say is a conditional statement.
Let p =It is raining, q =It is cloudy. Then the proposition can be written
as If p t
1.4 Arguments
Denition
A valid argument is a nite set of propositions P1 , ., Pr called premises,
together with a proposition C, the conclusion, such that the propositional
form (P1 P2 . Pr ) C is a tautology.
We say C follows logically from, or is a logi
2.2 Rules II
E (Eliminating the and)
pq
p and p q
q are both valid, so
If we have a step of the form p q in the proof, then we can
deduce p and we can deduce q.
I (Introducing and)
p, q
p q is valid, so
If we have steps of the form p and q in the proof, t
Solutions to Additional Questions 1
Those marked with a * may be considered harder than the rest.
1 Valid Arguments
Prove the following arguments are valid:
1. (A B) C, (A B)
C,
1 (A B) C
2 (A B)
3 C
2. (D E) (F G) , D E
A
A
DS 1,2
F G,
1 (D E) (F G) A
2