Up A Level
"And"
"And" of An "Or"
Contrapositive
"For All"
"If and Only If"
"If..., Then..."
"Not"
"Not" of An "And"
"Not" of An "If...Then"
"Not" of An "Or"
"Or"
"Or" of An "And"
Short Tautologies
"There exists"
| |
A Story of "And"
Let P and Q be sentences, each of which is either true or false (but neither is
both; there are weird sentences that are both---"this sentence is false").
"P and Q" is a compound sentence, which is true only when both
P and Q are true (and false in all other cases). Just to make it VERY
official, here is a table which lists every possible combination of truth
and falseness for P and Q and states whether "P and Q" is true for each possible
combination. I apologize for stating the obvious! In the tables, below,
T stands for true, and F stands for false.
Truth Table For "AND"
| P | Q |
P and Q |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
Consider one of the rows of this table, say the third. That row represents
this situation: P is false and Q is true; in this situation "P and Q" is
false. If you have any questions about this, please email ramsey@math.hawaii.edu with your question! It will be much appreciated.
What is the truth of "Q and P"? What difference does it make to "commute"
the two sentences P and Q by putting Q ahead of P? Having both P and Q true
has nothing to do with which is stated first (please forgive this review
of what may seem obvious). It makes no difference, which of P and Q is
stated first in an "and". "P and Q" MEANS EXACTLY THE SAME AS "Q and P";
the two compound sentences are true in exactly the same situations.
Again, to make it very official, we display this sameness of truth values
in the table below.
"AND" Is Commutative
| P | Q |
P and Q | Q and P |
| T | T | T | T |
| T | F | F | F |
| F | T | F | F |
| F | F | F | F |
What is the truth of "P and P"? If P is true, both parts of this "and"
sentence are true; by our understanding of "and", the compound "and"
sentence is true. If P is false, both parts of this "and" sentence are
false, and thus the compound "and" sentence is false. Thus, "P and P"
MEANS THE SAME THING AS "P". It may be redundant, even boring, but it's
true. Surprisingly, it is useful as well. Sometimes one wants to remove
redundancy to make shorter but equivalent sentences. Sometimes one wants
to introduce redundancy as part of a transformation into something that
IS interesting. The following table is intended to make this observation
quite official. Note that this table has only two rows because there are
only two situations to consider: P is true or P is false.
Redundancy with "AND"
| P | P |
P and P |
| T | T | T |
| F | F | F |
Consider, finally, an "and" sentence within an "and" sentence: "(P and Q)
and R". Here P, Q, and R are sentences (each one is either true or false,
but not both). The parentheses indicate that "P and Q" is compounded with
a third sentence R (think of "P and Q" being evaluated first for truthfulness,
and then R is taken into account). Inevitably, someone will create the
sentence "P and (Q and R)" [I just did]. By our understanding, of "and",
"(P and Q) and R" should mean the same as "P and (Q and R)"; in fact both
are true exactly when all three of P, Q, and R are true (and false in all
other situations). There are 8 situations to consider, T or F for each
of P, Q, and R in all possible ways. These are listed in the table below,
which lists the truth values of "(P and Q) and R".
"AND" Is Associative, part 1
| P | Q | R |
P and Q | (P and Q) and R |
| T | T | T | T | T |
| T | T | F | T | F |
| T | F | T | F | F |
| T | F | F | F | F |
| F | T | T | F | F |
| F | T | F | F | F |
| F | F | T | F | F |
| F | F | F | F | F |
Let us review one row of this table, say the fifth. There, P is false,
Q is true and R is true. Thus "P and Q" is false; from that, it follows
that "(P and Q) and R" is false because R is being "and"-ed with a
sentence which is false. Note that "(P and Q) and R" is true only in
the first situation, when all of P, Q, and R are true.
We now repeat this table for "P and (Q and R)":
"AND" Is Associative, part 2
| P | Q | R |
Q and R | P and (Q and R) |
| T | T | T | T | T |
| T | T | F | F | F |
| T | F | T | F | F |
| T | F | F | F | F |
| F | T | T | T | F |
| F | T | F | F | F |
| F | F | T | F | F |
| F | F | F | F | F |
Again, let us review the fifth row of this table. Because Q and R are
true there, "Q and R" is true; however, P is false; this makes
"P and (Q and R)" be false because "Q and R" is being "and"-ed with a
false sentence. Again, note that "P and (Q and R)" is true only in
the first situation when all three of P, Q and R are true.
Please email ramsey@math.hawaii.edu if you have questions about any of
the tables above.
Some Questions To Test Understanding.
- How many different compound sentences can be made from "(P and Q) and R"
by permuting the letters P, Q, and R? How many more can be made by permuting
the letters P, Q, and R and moving the parentheses to enclose the last two
sentences of the compound expression? Do the compound sentences created
by permutation and/or moving parentheses all mean the same thing? Try to
list all the possibilities in an order in which each one is obtained from
the previous one by one of two actions: commuting two sentences in an "and"
or by moving parenthesis.
- Consider four sentences "and"-ed: "(P and Q) and (R and S)". List
all the possible equivalent sentences obtained by moving parentheses and/or
permuting the letters P, Q, R and S. How many are there? Do they all mean
the same thing?
- Suppose that 27 sentences are "and"-ed in some order, with parentheses
placed so that each "and" connects only two sentences
(possibly compound themselves). When is this compound sentence true?
Some Uses of "And" in Writing About Mathematics
"And" appears frequently in this type of expression: a < b < c. This
really
means that a < b AND b < c. To repeat, b does both things:
b is bigger than
a AND b is less than c.
"And" can appear in proofs in one of two very ordinary ways. Suppose
that one has a paragraph that proves that some number x, say, is prime.
Then suppose that one has a second paragraph that shows that the same
x is bigger than
100. Then, without explanation, one can add a third paragraph consisting
of one sentence: "Thus, x is prime and x is bigger than 100." Or the
third paragraph can say: "Thus x is a prime which is bigger than 100."
Or "Thus, x is a prime bigger than 100." Often, there is no third paragraph.
The "and" of "x is prime" and "x is bigger than 100" may be used in later
paragraphs without explanation, because this use of "and" is thought to be
too obvious to mention.
Another use of "and" in proofs is equally ordinary. Suppose that you already
know that "P and Q" is true. Then you can use just P alone, or just Q alone,
without explanation. Both of them are true, and no justification is required
for using either of them. Here's an example. Suppose that you are told that x
is a perfect square strictly between 40 and 50 (strictly means not equal to
either 40 or 50). This is really an "and" sentence: (x is a perfect square)
and (40 < x < 50).
Here are some possible conclusions that writers of mathematics
might draw, without further explanation:
- There is some integer a such that x=a*a [that is, x is a times a].
What the writer has done here is grab the first part of the "and" (x is
a perfect square) and written the definition (which the writer assumes that
you already know) of what a perfect square is.
-
"x < 50". What the writer has done here is grab the second part of the
"and", "40 < x < 50". The writer assumes that you know "40 < x < 50"
is also an "and" sentence: "40 < x and x < 50". The writer then grabbed
the second
part of this second "and".
- "40 < x". See the previous example. This time, the writer has grabbed
the second sentence of the main "and", and then the first sentence of the
second "and" (the second "and" is in "40 < x < 50").
|