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Lecture 8, 7-August-20Discrete Mathematics 2003
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4.2 Propositions
In arithmetic we work with numbers Similarly, the fundamental
objects in logic
arepropositions
Definition: Aproposition is a statement
that is either true or false. Whichever of
these is the case is called the truth value of
the proposition.
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Examples of Propositions
1 There are 200 cents in a dollar.
2 The 1956 Olympic Games were held inMelbourne.
3 The assessment for Discrete Maths consists ofone exam and two
tests.
4 To pass Discrete Maths you must gain a markof at least 50% on
the exam.
5 Every even number greater than 2 is the sumof two prime
numbers.
(2) & (3) are true; (1) & (4) are false
(5) is either true or false noone knows which!
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Goldbachs Conjecture The proposition Every even number greater
than
2 is the sum of two prime numbers is known asGoldbachs
Conjecture
It appears to be true e.g. 20 = 7 + 13,54 = 17 + 37, 88 = ? + ?,
etc
However, noone has been able to show that everyeven no. is the
sum of 2 prime nos.
In 2000, there was even a $1 million prize (froma book
publisher) for a proof of the Conjectureby 15 March 2002 but nobody
claimed it
The Univ of Tennessee has some current
info:www.utm.edu/research/primes/notes/conjectures/
http://www.utm.edu/research/primes/notes/conjectures/http://www.utm.edu/research/primes/notes/conjectures/
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Lecture 8, 7-August-20Discrete Mathematics 2003
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Examples of non-Propositions
1 Stop talking.
2 What day of the week is it today?3 This sentence is false.
(1) is a command & (2) is a question soneither are
propositions
(3) is aself-referentialstatement i.e. itmakes a statement about
itself
(3) initially seems to be a proposition, but
problems occur when we try to determinewhether it is true or
false
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Paradoxes
This sentence is false is termed aparadox
Paradoxes cause difficulties if we allow them aspropositions so
we wont consider self-referential statements at all in our work on
logic
Sentences such as Winter is cold & Footballis boring are
sometimes not regarded aspropositions because it can be argued that
theirtruth values arent well defined
We willallow such sentences as propositions
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Predicates
y < 3 is an example of apredicate
A predicate is a statement which contains 1 or
more variables it cannot be assigned a truth
value until the value(s) of the variable(s) are
specified
e.g. ify = 2, the sentence is true, but ify = 4,
it is false
Well look at predicate logic later in Chap. 4
