(The numbers at the bottom of the table indicated the
order in which the columns were computed. Column 5
displays the possible truth-values of the statement p(q
→r) as a whole.) Thus if we deem pand rto be true and
qfalse, for example, then we must conclude that p(q
→r) as a as a whole is true. (Row three of the table.)
The truth table for p∨(
¬
p) is:
A compound statement that is true irrespective of the
truth-values of its component statements (that is, one
that has T for every row of its truth table) is called a
TAUTOLOGY
. Thus p∨(
¬
p), for example, is a tautol-
ogy. One can also check that (pq) →(r∨p) is
another example of a tautology.
A compound statement that is always false is called
a contradiction. For example, p(
¬
p) and [(p →q)
p] →(
¬
q) are contradictions.
Two compound statements are logically equivalent
if they have the same truth-values in the corresponding
rows of their truth tables. For example, one can check
that p→qand (
¬
p) ∨qhave identical truth tables and
are hence logically equivalent. The notion of such
equivalence is linked to the biconditional.
See also
ARGUMENT
;
LAWS OF THOUGHT
.
turning point A point on a differentiable graph y=
f(x) at which f′(x) = 0 and f′(x) changes sign is called a
turning point. Such a point is either a local maximum
or minimum for the graph.
See also
LOCAL MAXIMUM
/
LOCAL MINIMUM
.
twin primes Two consecutive odd numbers, both of
which are
PRIME
, are called twin primes. For example,
the pairs 3 and 5, 17 and 19, and 10,006,427 and
10,006,429 are twin primes. Mathematicians suspect
that there are infinitely many pairs of twin primes, but
are unable to prove this for a fact. This remains a
famous unsolved challenge.
The largest twin primes known, as of the year
2002, are 665551035 ×280025 – 1 and 665551035 ×
280025 + 1.
Two primes that differ by 4, such as 19 and 23 for
instance, are called cousin primes. There are other infor-
mal names for pairs of primes that differ by six, eight,
10, and 12. The name “twin primes” was coined by the
German mathematician Paul Stäckel (1892–1919).
The triple “3, 5, 7” is the only set of three consecu-
tive odd numbers that are each prime. (Any triple of
consecutive odd numbers must contain one number
divisible by 3. If the first odd number in the triple is
one more than a multiple of 3, then the second odd
number of the triple is a multiple of 3. If the first num-
ber in the triple is two more than a multiple of three,
then the third odd number is divisible by 3.)
two-card puzzle The following classic puzzle is
often used to demonstrate the need for care when per-
forming
PROBABILITY
calculations:
Two cards—one red on both sides, the other
red on one side and black on the reverse—are
placed into a bag. The bag is shaken, and one
card is removed. You see that one face of this
card is red. What is the probability that the
reverse side of that card is black?
One is tempted to answer one-half—after all, the
chances of selecting the red/black card is 1 in 2. How-
ever, this line of reasoning overlooks a subtlety: the puz-
zle is really one of
CONDITIONAL PROBABILITY
, and we
are being asked to determine the probability of selecting
the double-colored card given that the face we see is
red. As there are three red faces, only one of which con-
tains black on its reverse side, we conclude that there is
actually only a one-third chance of this being the dou-
ble-colored card. This surprising conclusion can be veri-
fied by performing the experiment multiple times.
See also B
AYES
’
S THEOREM
; K
RUSKAL
’
S COUNT
;
M
ONTY
H
ALL PROBLEM
.
∨
∨
∨
∨
∨
two-card puzzle 515
pp (
¬
p)
TTTF
FFTT
(1) (3) (2)
^