Parallelograms and Negating Sentences

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PARALLELOGRAMS and NEGATING SENTENCES

THEOREM: EQUIVALENT CHARACTERIZATIONS OF A PARALLELOGRAM

Let Q be a quadrilateral.
Then, the following are equivalent:

Q is a parallelogram (i.e., both pairs of opposite sides are parallel)
the diagonals of Q bisect each other
both pairs of opposite sides are equal
both pairs of opposite angles are equal
one pair of opposite sides are both parallel and equal

Use the Geometer's Sketchpad to create a quadrilateral where the diagonals bisect!
Use the Geometer's Sketchpad to create a quadrilateral where the opposite sides are equal!

You should be able to negate each of the statements in the theorem above,
and phrase these negations in a variety of ways.

Here are some logical skills for negating sentences.

How can a sentence A  and  B be false?
The only time that an "and" sentence is true is when both subsentences are true,
so an "and" sentence is false when at least one of the subsentences is false.

Precisely, the truth table below shows that
not(A  and  B) is equivalent to (not  A)  or  (not  B)

A	B	A and B	not(A and B)	not A	not B	(not A) or (not B)
T	T	T	F	F	F	F
T	F	F	T	F	T	T
F	T	F	T	T	F	T
F	F	F	T	T	T	T

How can a sentence A  or  B be false?
An "or" sentence is false only when both of the subsentences are false.

Precisely, the truth table below shows that
not(A  or  B) is equivalent to (not  A)  and  (not  B)

A	B	A or B	not(A or B)	not A	not B	(not A) and (not B)
T	T	T	F	F	F	F
T	F	T	F	F	T	F
F	T	T	F	T	F	F
F	F	F	T	T	T	T

Next, consider a sentence of the form For all  x,    P .
To understand a (true) sentence of this form, think of a population where each member has the property P , as shown below.

What does it mean to say that the sentence For all  x,    P is not true?
It means that at least one member of the population doesn't have property P, like this:

Of course, it could certainly be true that more than one member of the population doesn't have property P, like this:

Precisely,

not(  For all  x,    P  )
is equivalent to
There exists    x    such that  (not  P) .
Similarly,

not(  There exists    x    such that  P  )
is equivalent to
  For all  x,    (not  P  ) .

Now, we are ready to negate the statements in the "Equivalent Characterizations of a Parallelogram" theorem.

The negation of   Q is a parallelogram (i.e., both pairs of opposite sides are parallel)   is:
Q is not a parallelogram, or
it is not true that both pairs of opposite sides are parallel, or
at least one pair of opposite sides is not parallel , or
there exists a pair of opposite sides that is not parallel .

The negation of   the diagonals of Q bisect each other   is:
it is not true that the diagonals of Q bisect each other, or
at least one diagonal of Q is not bisected at the point of intersection of the diagonals , or
there exists a diagonal of Q that is not bisected at the point of intersection of the diagonals .

The negation of   both pairs of opposite sides are equal   is:
it is not true that both pairs of opposite sides are equal, or
at least one pair of opposite sides are not equal, or
there exists a pair of opposite sides that are not equal .

The negation of   both pairs of opposite angles are equal   is:
it is not true that both pairs of opposite angles are equal, or
at least one pair of opposite angles are not equal, or
there exists a pair of opposite angles that are not equal .

Notice that   one pair of opposite sides are both parallel and equal
can be re-phrased as
at least one pair of opposite sides are both parallel and equal or
there exists a pair of opposite sides that are both parallel and equal .

The negation of   one pair of opposite sides are both parallel and equal   is:
it is not true that one pair of opposite sides are both parallel and equal, or
each pair of opposite sides is either not parallel or not equal .

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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