An open note to Craig Laughton

Dear Craig,

In your journal entry dated 11/12/04 you express discomfort with the operator ≥. I'm not sure how you feel about it now, but just in case you're still unsure about it I thought I'd write this little note.

I think the first thing getting you into trouble is your pronunciation. ≥ is best pronounced 'at least'. The problem with `greater than or equal to' is that it leaves one with the mistaken impression that ≥ is a special case of > (as the commenter on your entry demonstrates). In fact it is better the other way round. We define > in terms of ≥ i.e.

x > y   ≡   x ≥ y  ∧  ¬(y ≥ x)

≥, of course has a counterpart, ≤, pronounced `at most' and you can find a nice little note about it here.

One property of ≥ is that it is reflexive i.e. x ≥ x for all x. Thus expressions like 14≥14 and 9≥9 are perfectly fine.

Regards,


Eric M.

The Σ calculus Part 4

Let us conclude (for now) our exploration of the Σ calculus by introducing
one new postulate and one more theorem.

Continued

The Σ calculus Part 3

Associativity and symmetry reveal themselves again in another postulate
- referred to by `interchange of quantifications'.

Continued

The Σ calculus Part 2

Formulae (3) and (4) relate quantifications with the same range. (4) has an analogue for different ranges.

Continued

The Σ calculus Part 1

In analogy to addition's symmetry and associativity, we postulate that summation distributes over addition.

Continued

The Σ calculus Part 0

Sums are common enough in programming that it is worth our time to become skilled at manipulating them. Here we give the repetoire of general formulae that are highly useful when dealing with sums.

Continued

Intermezzo

While working on a question I ran into a summation I did not know how to handle.
So I'd like to spend the next couple of posts looking at sums in general.
Today I'll start with an exercise from Concrete Mathematics. Knuth et al ask:

What does the notation

mean?

Continued

(1.2.1, 6)

Prove that if Eqs. (6) hold before step E4 is performed, they hold afterwards also.

Continued

Problem Proof 1 (1.2.1,3)

The following proof by induction seems correct, but for some reason the
equation for n=6 gives

on the left-hand side, and

on the right-hand side. Can you find a mistake?
Continued