Another form of the reciprocity law of Dedekind sum
... use the reciprocity law successively and also use the Matsumoto-Montesinos formula [7], which is proved by purely elementary number theoretic method (see also Remark 2.2). The proof of Proposition 1.2 is elementary, which is given in §3. Therefore the formula (3) is, in some sense, “another form” of ...
... use the reciprocity law successively and also use the Matsumoto-Montesinos formula [7], which is proved by purely elementary number theoretic method (see also Remark 2.2). The proof of Proposition 1.2 is elementary, which is given in §3. Therefore the formula (3) is, in some sense, “another form” of ...
Biform Theories in Chiron
... both about the syntax of expressions and what the expressions mean. Traditional logics are usually not equipped with the means to express statements about syntax and to reason about syntax. The transformer of a rule does not need to be expressed in the logic. As long as its corresponding meaning for ...
... both about the syntax of expressions and what the expressions mean. Traditional logics are usually not equipped with the means to express statements about syntax and to reason about syntax. The transformer of a rule does not need to be expressed in the logic. As long as its corresponding meaning for ...
Chapter 1, Part I: Propositional Logic
... “Inclusive Or” - In the sentence “Students who have taken CS202 or Math120 may take this class,” we assume that students need to have taken one of the prerequisites, but may have taken both. This is the meaning of disjunction. For p ∨q to be true, either one or both of p and q must be true. “Exclu ...
... “Inclusive Or” - In the sentence “Students who have taken CS202 or Math120 may take this class,” we assume that students need to have taken one of the prerequisites, but may have taken both. This is the meaning of disjunction. For p ∨q to be true, either one or both of p and q must be true. “Exclu ...
THE BINOMIAL THEOREM FOR HYPERCOMPLEX NUMBERS
... discovered quaternions in 1866. The algebra of quaternions is not commutative and therefore the usual binomial theorem fails there. We present a binomial theorem in H and even in a more general subspace Rn+1 of the Clifford algebra Cn . The (universal) Clifford algebra is the associative algebra over ...
... discovered quaternions in 1866. The algebra of quaternions is not commutative and therefore the usual binomial theorem fails there. We present a binomial theorem in H and even in a more general subspace Rn+1 of the Clifford algebra Cn . The (universal) Clifford algebra is the associative algebra over ...
Conjunctive normal form - Computer Science and Engineering
... functions, and propositional calculus—to compute the functional values of logical expressions on each of their functional arguments, that is, on each combination of values taken by their logical variables (Enderton, 2001). In particular, truth tables can be used to tell whether a propositional expre ...
... functions, and propositional calculus—to compute the functional values of logical expressions on each of their functional arguments, that is, on each combination of values taken by their logical variables (Enderton, 2001). In particular, truth tables can be used to tell whether a propositional expre ...
CS 40: Foundations of Computer Science
... c)The issue is ¬ e V d, which is equivalent to the conditional statement e → d. This does not follow from our assumptions. If we take d to be false, e to be true, and s to be false, then this proposition is false but our assumptions are true. d)The issue is ¬ d V ¬ e, which is equivalent to the cond ...
... c)The issue is ¬ e V d, which is equivalent to the conditional statement e → d. This does not follow from our assumptions. If we take d to be false, e to be true, and s to be false, then this proposition is false but our assumptions are true. d)The issue is ¬ d V ¬ e, which is equivalent to the cond ...
Practice Midterm 1
... (b) Suppose that 3 was a positive rational number. Let us say that 3 is equal to c/d, where c and d are natural numbers. Use part 4(a) to show that 3d − c < c. ...
... (b) Suppose that 3 was a positive rational number. Let us say that 3 is equal to c/d, where c and d are natural numbers. Use part 4(a) to show that 3d − c < c. ...