By Gabbay, D. M
Read or Download Fibred Semantics and the Weaving of Logics Part 1: Modal and Intuitionistic Logics [paper] PDF
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Additional info for Fibred Semantics and the Weaving of Logics Part 1: Modal and Intuitionistic Logics [paper]
Proof. It is given that sin(U ) = u/2v and from the definition of sine, sin(U ) = u/w, thus one has √ u/2v = u/w. By algebraic manipulations one obtains w = 2uv and hence the right triangle U V W is isosceles. ” Proof. The line L1 , being tangent to the endpoint of the diameter D of a circle, is perpendicular to D. Likewise, L2 is also perpendicular to D. As such, L1 and L2 are parallel. ” Fill in the details of any missing steps. CHAPTER 3: EXERCISES 39 √ Proof. It will be shown that w = 2uv, for then, by Proposition 3 on page 30, the right triangle U V W will be isosceles.
B. If n is a given integer satisfying −3n2 + 2n + 8 = 0, then 2n2 − 3n = −2. 15 List at least two key questions for each of the following problems. Be sure your questions contain no symbols or notation from the specific problem. a. If a and b are real numbers, then a2 + b2 ≤ (a + b)2 . b. If y = m1 x + b1 and y = m2 x + b2 are the equations of two lines for which m1 (m2 ) = −1, then the two lines are perpendicular. 16 List at least two key questions for each of the following problems. Be sure your questions contain no symbols or notation from the specific problem.
Do not forget to observe that the hypothesis of Proposition 1 also requires that the triangle RST be a right triangle, which of course it is, as stated in the current hypothesis. Notice how much more challenging it would be to match up the notation if the current triangle had been labeled W XY with sides of length w and x and hypotenuse of length y. This overlapping notation can (and will) arise and, when it does, you should use the same technique as in the case of definitions; that is, you should rewrite the previous proposition with a set of symbols that do not overlap with the current problem.