Inclusive or is a low hurdle. In plain English, when someone asks “Should we take our vacation to New York or Boston?” the assumption is that the answer will be “New York” or “Boston” (or “I don’t care” or “neither”). The geeky joke – sometimes serious – answer of “yes” is totally unhelpful. However, it’s not too hard to get used to inclusive or, and we do have examples in natural language. One of the best is “Would you like sugar or cream in your coffee?” Of course, even then “yes” isn’t a useful answer, since there are three possible coffee fixings that would lead to it.

Implication is a much higher bar; there’s still a part of me, even, that doesn’t think A implies B means much of anything when A is false. Implications where there is clearly no causal relationship between A and B can be helpful, since they rarely appear in plain English (outside of statistical correlations, I suppose) and thus resist natural language intuition. For teaching, in addition to that, I settled on the approach of “an implication is true unless proven false.” You can only prove that it’s false by having A be true and B be false, so in the A-false situations the implication is therefore true. This is basically the conversion of “A implies B” into the disjunction “B or not-A,” but hopefully in a way that doesn’t just shift the confusion to a different location.

I think the base of that confusion is a disconnect between allowed truth values. In plain English, a sentence can be true, false, or nonsensical. The third option is not permitted in mathematics (except in the sense of ill-formed formulas), and it is confusing that many implications that seem nonsensical or are constructed from false clauses (“if the moon is made of green cheese, then fish swim in the sea;” “if the moon is made of green cheese, then carriages turn into pumpkins at midnight”) are logically true.

In the fifteen-plus years since my bridge class, I have found only one plain English example of an implication considered true but constructed with false clauses, and in general my students were unfamiliar with it: the adage “If wishes were horses, beggars would ride.” I would love another, even though nowadays it would be purely for my own interest.

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e) Working only from the definitions of function, surjectivity, and injectivity, and not from other prior results, prove your answer for either line c) or line d) above. If there is an entry in the middle column, pin down the conditions under which that condition will hold of f.

No answers on this one, but a hint for part e): think of the function as a relation. That’s often helpful in working with functions between finite sets.

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Reflexivity is an existence property; a possession property. If you contain this entire particular set of pairs, you’re reflexive. If not, then not.

Symmetry and transitivity are implications; closure properties. Now some pairs don’t come for free – they require other pairs be added as well. Suppose you’re having a party. You have two friends who just started dating and are very attached – so if you have one of them over to the party you also have to have the other. That’s symmetry. You have another set of friends, a married couple with a new baby, who can’t get a babysitter. You can have just the husband or just the wife, but if you want to have both of them come, the baby has to come too. That’s transitivity.

In a numerical example: let A = {1,2,3,4,5}. Start building a relation R. Put (1,2) into R. If we want a symmetric R, we must also add (2,1). {(1,2)} is transitive, so for that we don’t need to add anything. However, if we put in (2,1) we must put in (1,1) and (2,2) for R to be transitive; this is called closing R under transitivity.

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Answer after the jump.

To succeed in this requires thinking of functions in the sense of “relations such that each element appears exactly once as the first element of a pair” and ignoring any pull toward functions as equations. The most general answer is to partition A, fix one element of each partition set, and map every element of that partition set to the fixed element. For example, map 1 and 2 both to 1, and map 3, 4, and 5 all to 4. Any constant function from A to A satisfies the requirements, at the size-1 partition end; the identity function arises from a partition of size 5. These are transitive because the only pairs of the form (a,b), (b,c) in the relation are those where b=c.

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There are many questions one can ask about counting: how many ways are there to fill a bag of chocolates (i.e., how many distinct lists of varieties might a bag contain)? How many ways to fill a bag include orange-filled chocolates? How many ways include both chocolate cream and mocha-filled chocolates? How many ways include neither vanilla cream nor raspberry-filled chocolates?

The question I’m more interested in discussing, however, is this: Suppose something is going wrong in the assembly line and over half the bags are coming out overfilled. Before you break open the chocolates and look at their insides, form a hypothesis as to what has gone wrong. What machine or machines are most likely to be malfunctioning? Why?

Your answer consists of your assumptions and the arguments based on them, with enough context restated from the problem that the answer reads smoothly.

There are a few conflicting but appropriate assumptions you could make here. The first is that either the malfunction is consistent, affecting all bags that come through that machine, or it’s inconsistent, affecting at most all of the bags, but possibly fewer.

The second choice of assumptions is whether the description “over half” implies “not all.” If every bag were coming out overfilled, you’d expect that to be specified… unless you’re a logician teaching a math class, in which case “over half” definitely includes the possibility of “all.”

But the important thing for writing up an answer is to be clear on what you’re assuming above and beyond the information given in the problem.

There is a third assumption that I expected everyone to make: that the most likely explanation involves malfunction of the fewest machines. It may not always be the case, but we’re assuming the candy factory is run well enough that machines are more likely to be working properly than improperly.

So what’s your answer?

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I’d have to be crazy to fall out of love with you.

Based on that hypothesis, he asserts

The place where I hold you is true, so I know I’m all right.

Is this a logically valid argument? That is, is still being in love a logically sound reason for Willie to assert he’s not crazy? Why or why not?

2. The King James translation of the Bible contains the following logically nontrivial statement:

Every one that hath forsaken houses, or brethren, or sisters, or father, or mother, or wife, or children, or lands, for my name’s sake, shall receive a hundredfold, and shall inherit everlasting life. [Matthew 19:29]

Write a sentence which gives the negation of that statement.

3. You are checking to see if it is always true that people who are taller than average dislike sitting in low chairs. In each of the following situations you have partial information. In which situations must you investigate further to make sure your premise is not invalidated, and what piece of information must you find out in each of those cases?
     a) The subject is 6′6″ tall.
     b) The subject is 4′11″ tall.
     c) The subject complains a lot about the low chair.
     d) The subject happily settles into the low chair.

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