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#1 2011-11-21 06:23:08

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True-false exercises in mathematics and other areas

For the true-false problems for this thread, all cases for a particular
statement must be true, else it is deemed false. That is, I am not
presenting exercises in the always true/sometimes true/never true format.

1. Within the confines of the United States, the westernmost part of Virginia
lies farther west than the westernmost part of West Virginia.

Suppose there are sets of fair, 6-sided dice, but they do not necessarily have
the values of 1 through 6 once each on each face.  Numbers other than
1 through 6 might be included, as well as certain sides could share the
same value.  An example might be a die with these values, one per each face:
2, 2, 3, 4, 5, 8.  The value of two or more dice thrown, such as A + A, is the sum
of the values of their top faces.

There are also sets of C dice and D dice with the above properties.
Dice X, Y, W, and Z are also like the above in properties, except that
they are nonspecific for the illustration below.

Given: One set of four fair, identically-numbered 6-sided dice and a different
set of four fair, identically-numbered dice.  X > Y means that die X
has a greater chance of having a larger value than die Y upon tossing them.
W < Z means that die W has a greater chance of having a smaller value
than die Z when the dice are tossed.

2. (i)

If A > B, and A + A  + A > B + B + B, then

A + A > B + B.

2. (ii)

If A + A < B + B and A + A + A + A < B + B + B + B, then

A + A + A < B + B + B

2. (iii)

If A + B > C + C, and A + B > D + D, then

A + B > C + D

"Lines" in the following refer to straight lines in the xy-plane.
Also, I am using "asymptote(s)" here to be only lines
(horizontal, vertical or oblique) instead of the nonlinear
asymptotes that exist for other certain functions.

3. (i)

Given:  A certain function approaches a line and gets ever closer
to it without intersecting it. 

That line is an asymptote to that function.

3. (ii)

A function does not interect its asymptote(s).

3. (iii)

The same function cannot have a horizontal asymptote
and an oblique asymptote.

4. In the vertical line test, where there is a graph of a relation
that is not a function, the vertical line crosses at two or more
points for each corresponding x-value in the relation's domain.

5. For any pair of perpendicular lines, the value of one slope
multiplied by the value of the other slope equals -1.

6. In plotting three points for a certain intended line (on graph paper,
for example) in the xy-plane, if a person has three points
that line up, then that person knows that those three points
lie on the same intended line.


8. There is a movie starring Jim Carrey and Kate Winslet
involving erasing of certain memories.

The title of this movie is one of the choices below, true or false:

"Spotless Mind of the Eternal Sunshine"

"Spotless Eternal Sunshine of the Mind"

"Spotless Sunshine of the Eternal Mind"

"Spotless Eternal Mind of the Sunshine"

"Eternal Spotless Mind of the Sunshine"

"Eternal Spotless Sunshine of the Mind"

"Eternal Mind of the Spotless Sunshine"

"Sunshine of the Spotless Eternal Mind"

"Sunshine of the Eternal Spotless Mind"

"Sunshine Eternal of the Spotless Mind"

"Sunshine Spotless of the Eternal Mind"

"Mind of the Eternal Spotless Sunshine"

"Mind of the Spotless Eternal Sunshine"

"Mind Spotless of the Eternal Sunshine"

"Mind Eternal of the Spotless Sunshine"

9.  In the same plane, two distinct quadrilaterals can intersect
each other in no more than eight distinct places.

10.  There are no infinite subsets (all members distinct) of the set
of irrational numbers such that they can be put into a one-to-one
correspondence with the set of positve integers.

11. Given: a, b, c, d belong to the Real numbers, where a, b, c, d > 1.

12.  A polyhedron exists that has seven edges.

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