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Math 135 Homework
Directions: Answer all the following questions. Be
sure you have read and thought about the material
in Chapter 1 in our text. Try to phrase your answers in complete sentences so
the reader has a narrative
instead of just short, clipped phrases or only symbolic notation . This is a
group homework set so you will
need to work with your group members to create solutions and write them up
nicely for presentation. Your
group'S written solutions are due Monday in class.
1) Give a definition for rational number . Using your
definition, prove that the sum, difference, product, and
quotient (with one exceptional case) of rational numbers are again rational
numbers. This is showing the
set of rational numbers is closed under these operations (again with one
exception in the case of division).
2) Consider these predicate symbols with the indicated
interpretations:
Rat(x) means `x is a rational number'.
Real(x) means `x is a real number '.
L(x; y) means `x is (strictly) less than y'.
a) Using these predicates, our two quantifier symbols,
variables, and logical connectives , translate this
statement (*) into our formal symbolism:
(*) On the real number line , between any two rational numbers there is another
rational number.
Informally speaking, this is saying the rational numbers are dense in
themselves.
b) Prove statement (*) using an appropriate proof method .
Consider this statement:
(**) On the real number line , between any two real numbers there is a rational
number.
c) Is (**) true or false? If false, produce a
counterexample; if true outline a proof strategy on how you might
prove it. What type of proof would you use, what context or basic assumptions
would be useful, etc?
3) Prove by contradiction that
is not rational. Note: this problem is NOT
directly connected to the
previous ones but certainly should use your definition of a rational number.
4) Rework this problem from a previous problem set:
a) Write out in our style of mathematical language the standard
definition of the following important
definition from calculus. Be sure to make all the quantifiers and logical
connectives explicit so we can see
the full logical structure of this statement.
Here we assume f is some specific function. Be sure you
have no free variables or mixed predicates and
quantifiers in your statement.
b) Now, form the negation of this statement and write it out in as positive a
form as you can.
c) Write up an explanation of how to use your negated statement to show the
following statement is false:
Notice that your statement in a) is universally quantified,
so its negation is existential. How do we show an
existential statement is true?
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