I see that I changed the sample inference
halfway through the hard-copy version
of the handout on the semantics of the predicate calculus,
switching the premise from ‘`P`(`t`(`c`))’
in the first section
to ‘`P`(`t`(`x`))’
in the third.
I have corrected this error in the on-line version of the handout,
with some adjustments both to the table and to the following text.

In today's class, I incautiously gave away the answer to exercise 4 of exercise set #3 even before the exercise set was due. So I'm replacing it with a different exercise, which will be due on November 21. (Any of you who still want to write up and submit a solution to the original problem may do so for extra credit.)

Here's the replacement exercise:

A *breakout* for a natural number `n`
is a bag of natural numbers that add up to `n`.
For example, the bag 〚 5, 1, 1, 0 〛
is a breakout of 7,
because 5 + 1 + 1 + 0 = 7.

Obviously, there are infinitely many breakouts of any natural number,
because any breakout can be extended by adding another 0 to the bag.
But there is still possible
to count breakouts of a specific natural number `n`
of a fixed bag cardinality `k`.
For example, there are exactly eleven four-member breakouts of 7,
namely
〚 7, 0, 0, 0 〛,
〚 6, 1, 0, 0 〛,
〚 5, 2, 0, 0 〛,
〚 5, 1, 1, 0 〛,
〚 4, 3, 0, 0 〛,
〚 4, 2, 1, 0 〛,
〚 4, 1, 1, 1 〛,
〚 3, 3, 1, 0 〛,
〚 3, 2, 2, 0 〛,
〚 3, 2, 1, 1 〛,
and 〚 2, 2, 2, 1 〛.

(a) For any natural numbers `n` and `k`,
let `K`(`n`, `k`) be the number
of `k`-member breakouts of `n`.
Write a recursive definition of the function `K`
and provide a combinatorial justification for your definition.

(b) Design, write, and test a Scheme procedure `breakouts`

that takes two natural numbers `n` and `k` as arguments
and returns a set containing all of the `k`-member breakouts of `n`.

(c) The shape

of a Skat hand is a bag
that contains the number of cards in each of the four suits in that hand.
For example, the shape of a Skat hand
that comprises five spades, one heart, three diamonds, and one club
is 〚 5, 1, 3, 1 〛.
Determine how many different shapes a Skat hand can have.

In case you need more practice
with formal proofs using the propositional calculus,
I've drawn up a list of important theorems
and put them in a handout:
Theorems of the Propositional Calculus.

Feel free to prove any of them that look appealing
and to use the ones that you prove
in subsequent proofs.

Here are the links to the readings for Monday's class:

Axiom Systems and Mathematical Theories

Bags

Bell Numbers and Integer Partitions

Definitions and Libraries

Factorials, Falling Powers, and Binomial Coefficients

Laws of Relations

Laws of Sets

Permutations and Combinations

The Predicate Calculus

The Propositional Calculus

Records in R

^{7}RS SchemeScheme: A Refresher

Scheme in DrRacket

Semantics of the Predicate Calculus

Stirling Numbers

Text Formatting and Typesetting with L

^{A}T_{E}XTheorems of the Propositional Calculus

Truth Tables

Evaluating Sentences of the Predicate Calculus in Finite Universes

Generating and Counting Bags

Generating Permutations and Combinations

Getting Started with L

^{A}T_{E}XLibraries in R

^{7}RS SchemeProse Proofs about Sets

Proofs about Sets

Proofs in the Predicate Calculus

Records in R

^{7}RS SchemeSetting up DrRacket for R

^{7}RS ProgrammingTruth-Table Semantics

Using the

/`(discrete sets)`

Library