Applying Logical Equivalences to Algebraic and Geometric Statements

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Applying Logical Equivalences to Algebraic and Geometric Statements

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A postulate is a statement that is assumed to be true without proof.
A theorem is a statement that is true, important, and has been proved.

This section presents some basic postulates of geometry and algebra.
You will use these postulates, and other mathematical sentences,
to practice working with logical equivalences.

Consider this true sentence:
(*) All squares are rectangles.

UNIVERSAL SET:
Square and rectangles are both members of the larger population of quadrilaterals.

IMPLICATION FORM of sentence (*):
(**) For all quadrilaterals Q,
if Q is a square, then Q is a rectangle.

Use this logical equivalence ...

... to rewrite sentence (**) as ...

If A then B

is equivalent to

If (not B) then (not A)

For all quadrilaterals Q,
if Q is not a rectangle, then Q is not a square.

If A then B

is equivalent to

(not A) or B

For all quadrilaterals Q,
Q is not a square, or Q is a rectangle.

Note: "For all" sentences will be studied in depth in a later section.

Geometry Postulate:
(*) Two distinct points determine a line.
Note: distinct means different

UNIVERSAL SET:
the set of all points

IMPLICATION FORM of sentence (*):
(**) For all points P and Q ,
if P and Q are distinct, then P and Q determine a unique line.

THE IDEA:

two distinct points determine a unique line

non-distinct points determine infinitely many lines

Use this logical equivalence ...

... to rewrite sentence (**) as ...

If A then B

is equivalent to

If (not B) then (not A)

For all points P and Q ,
if P and Q do not determine a unique line,
then P and Q are not distinct.

If A then B

is equivalent to

(not A) or B

For all points P and Q ,
P and Q are not distinct,
or P and Q determine a unique line.

Algebraic Postulate: the ZERO FACTOR LAW
(*) If two numbers multiply to zero, then at least one of the numbers is zero.

UNIVERSAL SET:
the set of all real numbers

MATHEMATICAL FORM of sentence (*):
(**) For all real numbers a and b ,
if ab=0 then ((a=0) or (b=0))

THE IDEA:
I'm thinking of two numbers that multiply to zero.
Can you tell me anything about the numbers that I'm thinking of?
Yes! At least one of them must equal zero.

Use these logical equivalences ...

... to rewrite sentence (**) as ...

If A then B

is equivalent to

If (not B) then (not A)

not(A or B)

is equivalent to

(not A) and (not B)

For all real numbers a and b ,
if a≠0 and b≠0 ,
then ab≠0 .

If A then B

is equivalent to

(not A) or B

(A or B) or C

is equivalent to

A or (B or C)

For all real numbers a and b ,
ab≠0 or a=0 or b=0 .

The table below summarizes more results that you need to know,
and that will be used to practice the ideas in this section:

NAME

COMMENTS

Geometry Postulate:
Three noncollinear points determine a unique plane.

three collinear points determine infinitely many planes

Substitution
If a=b , then a and b can be substituted, one for the other, in any expression.

Numbers have lots of different names!
Use whatever name is most convenient in a given situation.

Addition Property of Equality
For all real numbers a , b , and c ,

a=b   ⇔  a+c=b+c   .

This postulate gives you something that you can do to an equation that will always yield an equivalent equation.

The Addition Property of Equality says that you can
add (or subtract) the same number to (or from) both sides of an equation,
and it will not change the truth of the equation.
Read about the Addition Property of Equality in:
One Mathematical Cat, Please! A First Course in Algebra

Multiplication Property of Equality
For all real numbers a and b , and for c&neq;0 ,

a=b   ⇔  a⁢c=b⁢c   .

This postulate gives you something that you can do to an equation that will always yield an equivalent equation.

The Multiplication Property of Equality says that you can
multiply (or divide) both sides of an equation by the same nonzero number,
and it will not change the truth of the equation.
Notice that multiplying both sides of an equation by zero can change the truth of the equation:
" 3 = 5 " is false, but " 3·0 = 5·0 " is true.
Read about the Multiplication Property of Equality in:
One Mathematical Cat, Please! A First Course in Algebra

NEGATING EQUATIONS and INEQUALITIES:

STATEMENT NEGATION

a=b a≠b

a<b a≥b

a>b a≤b

You must be able to negate each of these sentences in the specified form.

NEGATING AN IMPLICATION:

not(A⇒B)

is equivalent to

A and (not B)

An implication is false when its hypothesis is true and its conclusion is false.

Finally, never lose sight of the following key facts!

Two sentences are equivalent if they always have the same truth values:
if one is true, so is the other; if one is false, so is the other.
Sentences that are equivalent can be used interchangeably, and a mathematician will use whichever is easiest.
Equivalence is such an important mathematical concept that there are many ways to say the same thing.
The following four sentences are completely interchangeable:

A  is equivalent to  B
A  if and only if  B
A  iff  B
A  ⇔  B

On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.

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