Problem 1 asks to show that an integer is odd if and only if it is the sum of two consecutive integers.

Here is an argument: Let be an integer. We need to prove two statements, corresponding to the two directions of the “if and only if:”

If is odd, then there are two consecutive integers and such that

If is the sum of two consecutive integers and then is odd.

1. Assume that is odd. Then, by definition, there is an integer such that Note that so we can take and we see that and are consecutive, and

2. Now assume that where and are consecutive integers. Say, Then By definition, this means that is odd.

Problem 2 asks to show that if are integers and we have that and then also

To see this, assume that are integers such that and By definition, this means that there are integers and such that and Then But Let Then is an integer, and By definition, this means that

Problem 3 asks to show that if is a positive integer, then is composite.

As stated, this is false. For a counterexample, note that is a positive integer. However, is not composite according to the definition given in the book:

An integer is composite if and only if there is an integer such that and

In any case, it is true that if is an integer and then is composite. To see this, note first that Now, if then and so Also, since then and therefore We have shown that and that By definition, this means that is composite.

Advertisements

Like this:

LikeLoading...

Related

This entry was posted on Monday, February 8th, 2010 at 8:28 pm and is filed under 187: Discrete mathematics. You can follow any responses to this entry through the RSS 2.0 feed.
You can leave a response, or trackback from your own site.

The only reference I know for precisely these matters is the handbook chapter MR2768702. Koellner, Peter; Woodin, W. Hugh. Large cardinals from determinacy. In Handbook of set theory. Vols. 1, 2, 3, 1951–2119, Springer, Dordrecht, 2010. (Particularly, section 7.) For closely related topics, see also the work of Yong Cheng (and of Cheng and Schindler) on Harr […]

As other answers point out, yes, one needs choice. The popular/natural examples of models of ZF+DC where all sets of reals are measurable are models of determinacy, and Solovay's model. They are related in deep ways, actually, through large cardinals. (Under enough large cardinals, $L({\mathbb R})$ of $V$ is a model of determinacy and (something stronge […]

Throughout the question, we only consider primes of the form $3k+1$. A reference for cubic reciprocity is Ireland & Rosen's A Classical Introduction to Modern Number Theory. How can I count the relative density of those $p$ (of the form $3k+1$) such that the equation $2=3x^3$ has no solutions modulo $p$? Really, even pointers on how to say anything […]

(1) Patrick Dehornoy gave a nice talk at the Séminaire Bourbaki explaining Hugh Woodin's approach. It omits many technical details, so you may want to look at it before looking again at the Notices papers. I think looking at those slides and then at the Notices articles gives a reasonable picture of what the approach is and what kind of problems remain […]

It is not possible to provide an explicit expression for a non-linear solution. The reason is that (it is a folklore result that) an additive $f:{\mathbb R}\to{\mathbb R}$ is linear iff it is measurable. (This result can be found in a variety of places, it is a standard exercise in measure theory books. As of this writing, there is a short proof here (Intern […]

Very briefly: Yes, there are several programs being developed that can be understood as pursuing new axioms for set theory. For the question itself of whether pursuing new axioms is a reasonably line of inquiry, see the following (in particular, the paper by John Steel): MR1814122 (2002a:03007). Feferman, Solomon; Friedman, Harvey M.; Maddy, Penelope; Steel, […]

This is a very interesting question and the subject of current research in set theory. There are, however, some caveats. Say that a set of reals is $\aleph_1$-dense if and only if it meets each interval in exactly $\aleph_1$-many points. It is easy to see that such sets exist, have size $\aleph_1$, and in fact, if $A$ is $\aleph_1$-dense, then between any tw […]

Say that the triangle is $ABC$. The vector giving the median from $A$ to $BC$ is $(AC+AB)/2$. Similarly, the one from $B$ to $AC$ is $(BA+BC)/2$, and the one from $C$ to $BA$ is $(CB+CA)/2$. Adding these, we get zero since $CB=-BC$, etc.

The usual definition of a series of nonnegative terms is as the supremum of the sums over finite subsets of the index set, $$\sum_{i\in I} x_i=\sup\biggl\{\sum_{j\in J}x_j:J\subseteq I\mbox{ is finite}\biggr\}.$$ (Note this definition does not quite work in general for series of positive and negative terms.) The point then is that is $a< x