MthT 491 Mathematical Statements
MthT 491 Mathematical Statements




Formulation of Mathematical Propositions


The following observations are motivated by the discussion in Chapter 1 of An Introduction to the Theory of Numbers by Ivan Niven and Herbert Zuckerman .
Students (including me!) are often tripped by mathematical statements which are stated differently as a matter of convenience or style. To quote :


... if A denotes some assertion or collection of assertions, and B likewise, the following statements are equivalent - they are just different ways of saying the same thing.
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A implies B.
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If A is true, then B is true.
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In order that A be true it is necessary that B be true.
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B is a necessary condition for A.
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A is a sufficient condition for B.
We add the equivalent statements:

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If A, then B.
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Whenever A is true, B is true.
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Whenever A, B.
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A implies B.
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B is implied by A.
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A ⇒ B.
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Satisfying A implies satisfying B.
Equivalent Mathematical Statements
1.
Division and Quotients
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18 divided by 6 is 3.
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18 divided by 6 equals 3.
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18/6 = 3.
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[18/6] = 3.
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18 ÷6 = 3.
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18 over 6 is 3.
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….
2.
Inequalities and Positives
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a is less then b.
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a < b.
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b > a.
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b − a is positive.
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a − b is negative.
3.
Multiples and Factors
(suggested by F. L. Porter)
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2 is a factor of 10.
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10 is a multiple of 2.
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2 divides 10.
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2 |10.
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10 is divisible by 2.
Mathematical Statements
The following is almost entirely from Peter J. Eccles, An Introduction to Mathematical Reasoning [E].
So what is a statement ?
First we consider a proposition . According to Eccles, a proposition is a sentence which is either true or false (but not both) . Examples of propositions (Eccles, p. 3):
1.
1 + 1 =2. True
2.
π = 3. False
3.
12 may be written as the sum of two prime numbers. True
4.
The square of an even integer is even. True
The following are not propositions:
5.
12 − 11. Not a sentence
6.
n is a prime number. True for some n, False for some n. Can be made a proposition if a particular value or condition is assigned to the variable n.
7.
m2 − 2n > 0. True for some n, m. False for some m,n. Can be made a proposition if particular values or conditions are assigned to m,n.
A sentence which can be made into a proposition when values are assigned to certain free variables is called a predicate .
Whether a sentence is a proposition depends on the context - are we speaking about positive integers, just positive integers, real numbers, …? To quote Eccles:

The fact that a sentence is a proposition relies on a number of assumptions about the meanings of the symbols and words which have been taken for granted.

Definition. A [mathematical] statement is a sentence which is a proposition or a predicate.


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On 15 Sep 2014, 20:40.