Question
In this question, you will prove that every truth table can be implemented us- ing only the conditional operator and the constant False . Hint:
In this question, you will prove that every truth table can be implemented us-
ing
only
the conditional operator
and the constant
False
. Hint: in your justifications,
you are free to use the fact that
a
b
a
b
.
a.
[3 marks]
Show that
can be simulated using only
and
False
. That is, design
a logical expression that is logically equivalent to
p
, and uses only
p
,
and
False
.
You must prove that your logical expression is correct using equivalence rules. (See,
e.g., Epp 4th ed page 35, Epp 5th ed page 49, or Daves excellent formula sheet)
b.
[3 marks]
Show that
can be simulated using only
and
False
. That is, design a
logical expression that is logically equivalent to
p
q
and uses only
p
,
q
,
and
False
.
You must prove that your logical expression is correct using equivalence rules.
c.
[3 marks]
Show that
can be simulated using only
and
False
. That is, design a
logical expression that is logically equivalent to
p
q
and uses only
p
,
q
,
and
False
.
You must prove that your logical expression is correct using equivalence rules.
Since for every truth table over
k
atomic propositions, we can write a propositional form
that matches the truth table using
,
and
, your answers to parts (a), (b) and (c)
show that you every propositional form is logically equivalent to a propositional form that
uses only
and
False
.
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