1. Glossary: Ex. 1
Solution .
1. A statement.
2. A statement.
3. A statement. Statements do not have to be true.
4. A statement. We do not have to be able to verify the truth of a statement.
5. Not a statement. A question cannot be a statement.
6. Not a statement. This is not a declarative sentence.
7. Not a statement. A statement is either true or false. This sentence is a
paradox: if it
is true, then by what it says, it is false, but then its negation is that "This
sentence is
true", etc.
8. Not a statement. This is not a meaningful sentence.
2. The Propositional Calculus: Ex. 2
Solution.
•If f is a differentiable function , then f is continuous.
•If α and β are two right
angles, then α = β .
•If M is a matrix with a zero eigenvalue , then M is not invertible.
Table 1: Truth table for The Propositional Calculus: Ex.
7(a)
Table 2: Truth table for The Propositional Calculus: Ex.
7(e)
Table 3: Truth table for The Propositional Calculus: Ex.
7(f)
Table 4: Truth table for The Propositional Calculus: Ex.
7(h)
3. So low : Ex. 2.9
(a) How can I show that one real number is less than or
equal to another real number?
How can I show that the sum of the squares of two non- negative real numbers is
less
than or equal to than the square of their sum ?
b) How can I show that two lines are parallel ? How can I
show that two lines do not
intersect? How can I show that if two lines have equations in `ax + b' form with
the
same x- coefficient , then their slopes are the same ?
4. Solow: Ex. 2.21
Solution. Let A be the statement: RST is a triangle
such that SU is a perpendicular
bisector of RT. Let B be the statement: triangle SUR is congruent to triangle
SUT. We
are trying to show that A -> B.
A key question is: how can I show that two triangles are
congruent? Moving forward from
A, because SU is a perpendicular bisector of RT, we obtain statement A1:
Also moving forward from A, because SU is a perpendicular bisector we obtain
statement
A2: 
. An answer to the key question is: show
that two sides of
the triangles and the included angle are equal. A specific answer is: show that
is clear, so we are done.
5. Solow: Ex. 3.14
Analysis of Proof. The forward-backward method
gives rise to the key question "How
can I show that a real number is rational?". One answer is to use the definition
of a rational
number as a number which can be ex pressed as a ratio of integers with nonzero
denominator.
So we must show that
B1 : a + b can be expressed as a ratio of integers
with nonzero denominator.
Turning to the forward process, we can use the definition
of a rational number to obtain
statements
A1 :
where p and
q are integers and q ≠ 0
and
A2 : 
where r
and s are integers and s ≠ 0.
Hence
Continuing forward from this, by carrying out the addition
of fractions we obtain
Returning to the backward process, it is now enough to
establish statement
B2 : 
is a ratio
of integers with nonzero denominator.
Going forward again, combining the facts about p, q, r, s
in statements A1 and A2, we obtain
A5 : ps + rq is an integer.
and
A6 : qs is a nonzero integer.
Statements A5 and A6 together establish
statement B2, and the proof is complete.
Proof. By the definition of rational numbers, since a is a
rational number, there are integers
p and q, with q ≠ 0, such that 
Similarly, there are integers r and s, with s ≠ 0, such
that 
We now compute:
Now ps + rq and qs are both integers, since p, q, r and s
are all integers. Moreover, qs ≠ 0
since q and s are both nonzero. Hence a + b can be expressed as the ratio of two
integers
such that the denominator is not zero . We conclude that a + b is rational.
