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During this booklet Nijenhuis and Wilf speak about a variety of combinatorial algorithms.
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Extra resources for Combinatorial Chemistry and Technology
The third and fourth rows of the truth table, which say that the statement P → Q is true whenever P is false, regardless of the value of Q, are less intuitively obvious. There is, however, no other plausible way to fill in these rows, given that we want the entries in the truth table to depend only on the truth or falsity of P and Q, and that the one situation with which we are primarily concerned is that we do not want P to be true and Q to be false. Moreover, if we were to make the value of P → Q false in the third and fourth rows, we would obtain a truth table that is identical to the truth table for P ∧ Q, which would make P → Q redundant.
What would it mean to say that this statement is true? It would not mean that Fred is going on vacation, nor would it mean that Fred will read a book. The truth of this statement means only that if one thing happens (namely, Fred goes on vacation), then another thing will happen (namely, Fred reads a book). In other words, the one way in which this statement would be false would be if Fred goes on vacation, but does not read a book. The truth of this statement would not say anything about whether Fred will or will not go on vacation, nor would it say anything about what will happen if Fred does not go on vacation.
This argument is invalid, which we can see as follows. ” The argument then becomes A→R S→H A∧H R ∧ S. Suppose that A is true, that R is true, that S is false and that H is true. Then A → R and S → H and A ∧ H are all true, but R ∧ S is false. Therefore the premises are all true but the conclusion is false, which means that the argument is invalid. For some other combinations of A, R, S and H being true or false, it works out that the premises are all true and the conclusion is true, and for some combinations of A, R, S and H being true or false, it works out that the premises are not all true (in which case it does not matter whether the conclusion is true or false for the conclusion to be implied by the premises).