MATHEMATICAL LANGUAGE AND
SYMBOLS
“Like any language, mathematics has its own syntax, and rules.”
Learning the Language of Mathematics
(Jamison, 2000)
Unlike the language of ordinary speech, mathematical language is
nontemporal
devoid of emotional content
precise
Example: The word “any” in ordinary speech is ambiguous.
Can anyone work this problem?
(existential qualifier)
Anyone can do it!
(universal qualifier)
Definitions
(Jamison, 2000)
A definition is a concise statement of the basic properties of an object or
concept which unambiguously identify that object or concept.
Every concept is defined as a subclass of a more general concept called
genus
.
Each special subclass of the genus is characterized by special features called
the
species
.
Good Definition
A rectangle is a
quadrilateral
all four of whose angles are right angles.
Definitions
(Jamison, 2000)
Poor Definition
(not concise)
A rectangle is a parallelogram in which the diagonals have the same length and all the angles are
right angles. It can be inscribed in a circle and its area is given by the product of two adjacent
sides.
Poor Definition
(not basic)
A rectangle is a parallelogram whose diagonals have equal lengths.
Bad Definition
(ambiguous)
A rectangle is a quadrilateral with right angles.
Unacceptable Definition
(no genus)
Rectangle: has right angles
Important Statements
1.
A universal Statements says that a certain property is true for all
elements in a set.
Example:
All positive numbers are greater than zero.
2. A conditional statements says that if one thing is true then some other
thing also has to be true.
Example:
If 378 is divisible by 18, then 378 is divisible by 6.
3. Given a property that may or may not be true, an Existential
Statement says that there is at least one thing for which the property
is true.
Example:
There is a prime number that is even.
Universal Conditional Statements




Universal statements contain variation of the words “for all “ and conditional statements
contain versions of the words ”if
–
then”.
A universal conditional statement is a statement that is both universal and conditional
Example:
For all animals
a
, if
a
is a dog, then
a
is a mammal.
Rewriting this:
If a is a dog, then a is a mammal.
If an animal is a dog, then the animal is a mammal.
For all dog a, a is a mammal.
All dogs are mammal.
Let’s try this…
a.
b.
c.
d.
e.
Fill in the blanks to rewrite the following statement.
For all real numbers x, if x is nonzero then x
2
is positive
If a real number is nonzero, then its square _____________.
For all nonzero real numbers x, ____________.
If x ____________, then _____________.
The square of any nonzero real number is ______________.
All nonzero real numbers have ________________.
Answer:
a.
b.
c.
d.
e.
For all real numbers x, if x is nonzero then x
2
is positive
If a real number is nonzero, then its square
is positive.
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 Winter '17