MIT6_042JF10_rec01_sol - 6.042/18.062J Mathematics for...

6.042/18.062J Mathematics for Computer Science September 10, 2010
Tom Leighton and Marten van Dijk
Notes for Recitation 1
1 Logic
How can one discuss mathematics with logical precision, when the English language is itself
riddled with ambiguities? For example, imagine that you ask a friend what kind of dessert
was offered at the party you couldn’t make it to last week, and your friend says,
You could have cake or ice cream.
Does this mean that you could have both cake and ice cream? Or does it mean you had to
choose either one or the other?
To cope with such ambiguities, mathematicians have defined precise meanings for some
key words and phrases. Furthermore, they have devised symbols to represent those words.
For example, if P is a proposition, then “not P ” is a new proposition that is true whenever
P is false and vice versa. The symbolic representation for “not P ” is P or P . ¬
Two propositions, P and Q , can be joined by “and”, “or”, “implies”, or “if and only if”
to form a new proposition. The truth of this new proposition is determined by the truth of
P and Q according to the table below. Symbolic equivalents are given in parentheses.
“ P implies Q ” or “ P if and only if Q ” or
“ P and Q ” “ P or Q ” “if P , then Q ” “ P iff Q ”
P
F
Q
F
( P ∧ Q )
F
( P ∨ Q )
F
( P ⇒ Q )
T
( P ⇔ Q )
T
F T F T T F
T F F T F F
T T T T T T
There are a couple notable features hidden in this table:
• The phrase “ P or Q ” is true if P is true, Q is true, or both. Thus, you can have your
cake and ice cream too.
• The phrase “ P implies Q ” (equivalently, “if P , then Q ”) is true when P is false or
Q is true . Thus, “if the moon is made of green cheese, then there will be no final in
6.042” is a true statement.

� � � �
Recitation 1 2
There are two more important phrases in mathematical writing: “for all” (symbolized
by ∀ ) and “there exists” (symbolized by ∃ ). These are called quantifiers . A quantifier is
always followed by a variable (and perhaps an indication of the range of that variable) and
then a predicate, which typically involves that variable. Here are two examples:
x
∀ x ∈ R
+
e < (1 + x )
1+ x
∃ n ∈ N 2
n
> (100 n )
100
x The first statement says that e is less than (1 + x ) 1+ x for every positive real number x .
The second statement says that there exists a natural number n such that 2 n > (100 n ) 100 .
The special symbols such as ∀ , ∃ , ¬ , and ∨ are useful to logicians trying to express
mathematical ideas without resorting to English at all. And other mathematicians often
use these symbols as a shorthand. We recommend using them sparingly, however, because
decrypting statements written in this symbolic language can be challenging!