Definition: Contradiction
Some compound statements are always false. Such a statement is called a
contradiction
.

Study unit 9
Logic:
Truth tables
COS1501/1
146
Activity 9-6: Self-assessment exercises
1.
Express the following sentence symbolically and then determine
whether or not it is a tautology:
If demand has remained constant and prices have been increased, then
turnover must have decreased.
Use p for “demand has remained constant”, q for “prices have been
increased” and r for “turnover must have decreased”.
2.
Refer to Activity 9-5, Question 2. From the truth tables you have
constructed for (a) to (g), determine whether each of the statements is a
tautology, a contradiction or neither of the two.
Suppose that a and b are statements, and not necessarily simple ones. Then we can use
the concept of tautology to spell out the idea that a and b have the same meaning or that
a and b say the same thing in different words.
Definition: Logical equivalence
The two statements a and b are
logically equivalent
, denoted by a
{
b, if and
only if the statement a
↔
b is a tautology.
Activity 9-7: Truth table for the biconditional
Recall that a
↔
b has the value T if and only if a and b have the same truth
value. So, to check that a
↔
b is always T, it is enough to check that the final
columns in the truth tables of a and b are identical.
Example
Let’s look at the truth table for (p
→
q)
↔
(¬ q
→
¬ p):
p
q
¬ q
¬ p
p
→
q
¬ q
→
¬ p
(p
→
q)
↔
(¬ q
→
¬ p)
T
T
F
F
T
T
T
T
F
T
F
F
F
T
F
T
F
T
T
T
T
F
F
T
T
T
T
T
Because there are only T’s in the final column, it follows that
(p
→
q)
↔
(¬ q
→
¬ p) is a tautology.
This tells us that p
→
q and ¬ q
→
¬ p are logically equivalent,
i.e.
p
→
q
{
¬ q
→
¬ p
i.e.
p
→
q has exactly the same meaning as ¬ q
→
¬ p.
Note that
{
is
not
just another way to write
↔
.
We may write p
→
q
{
¬ q
→
¬ p only because we have shown that
(p
→
q)
↔
(¬ q
→
¬ p) is a tautology.

Study unit 9
Logic:
Truth tables
COS1501/1
147
Activity 9-8: Important logical equivalences
Take note of the following important logical equivalencies:
(a)
p
∨
q
{
q
∨
p
p
∧
q
{
q
∧
p
(commutative laws)
(b)
p
∨
(q
∨
r)
{
(p
∨
q)
∨
r
p
∧
(q
∧
r)
{
(p
∧
q)
∧
r
(associative laws)
(c)
p
∧
(q
∨
r)
{
(p
∧
q)
∨
(p
∧
r)
p
∨
(q
∧
r)
{
(p
∨
q)
∧
(p
∨
r)
(distributive laws)
(d)
p
∨
p
{
p
p
∧
p
{
p
(idempotent laws)
(e)
¬ (¬ p )
{
p
(law of double negation)
(f)
¬ (p
∨
q)
{
¬ p
∧
¬ q
¬ (p
∧
q)
{
¬ p
∨
¬ q
(De Morgan’s laws)
(g)
p
∨
¬ p
{
T
O
, where T
O
is a tautology
p
∧
¬ p
{
F
O
, where F
O
is a contradiction (negation)
(h)
¬ F
O
{
T
O
¬ T
O
{
F
0
(negations of T
O
and F
0
)
(i)
p
∨
F
O
{
p
p
∧
T
O
{
p
(identity)
(j)
p
∨
T
O
{
T
O
p
∧
F
O
{
F
O
(universal bound)
We often refer to these as
identities.
(You can use truth tables to verify that these are indeed logical equivalences.)
Now that we have the notion of logical equivalence, we can derive a rather surprising
result: We only require negation plus the connectives
∧
and
∨
!

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