Let domain of discourse be the integers, and let n and c be nonzero integers. Consider...
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Let domain of discourse be the integers, and let n and c be nonzero integers. Consider the following claim: Va Vb ((ca cn cb) → (a =n b)) This last "" is actually an "", but to make the problem easier, we have only asked you to prove "→". (a) [1 Point] Translate the claim into English. (b) [12 Points] Write a formal proof that the claim holds. Some important notes: Cozy has special notation for the predicate "=", but for everything else, it uses predicate notation. In particular, a b is written Divides (a, b), and a = b is written Congruent (a, b, m) ■ You will need to make use of Cozy's cite rule, which cite's a known theorem. In this case, the theorem you want to use is DivideEqn, which says the following about integers: " Va VbVc ((ca = cb) A(c =0)) + (a = b)) Cozy's apply rule is an even easier way to cite and use a Theorem. See the Cozy documentation for an explanation of how to use apply. Cozy can only apply DivideEqn to an equation that looks exactly like c(...) = c(...). For example, it cannot be applied to the equation c = ca+cb. Instead, you would first rewrite it as c.1 = c(a+b) using the algebra rule and then apply DivideEqn. In particular, note that cab means (ca)b in Cozy because multiplication associates to the left, so you would need to explicitly transform cab = cde to c(ab) = c(de) using algeaba before you can divide both sides by c. ■ Remember that Cozy expects a "*" for multiplication. It will misunderstand ca = cb. You have to write that as c⭑a = c*b in Cozy, even though we would write it as ca = cb for a human reader. Submit and check your formal proof here: http://cozy.cs.washington.edu You can make as many attempts as needed to find a correct answer. (c) [3 Points] Translate your proof to English. This week, your English proofs will be graded not just for effort, but also for correctness. Keep in mind that your proof will be read by a human, so you need to explain the algebra steps in more detail than you would with Cozy. (Follow the same rules as in Problem 1.) Let domain of discourse be the integers, and let n and c be nonzero integers. Consider the following claim: Va Vb ((ca cn cb) → (a =n b)) This last "" is actually an "", but to make the problem easier, we have only asked you to prove "→". (a) [1 Point] Translate the claim into English. (b) [12 Points] Write a formal proof that the claim holds. Some important notes: Cozy has special notation for the predicate "=", but for everything else, it uses predicate notation. In particular, a b is written Divides (a, b), and a = b is written Congruent (a, b, m) ■ You will need to make use of Cozy's cite rule, which cite's a known theorem. In this case, the theorem you want to use is DivideEqn, which says the following about integers: " Va VbVc ((ca = cb) A(c =0)) + (a = b)) Cozy's apply rule is an even easier way to cite and use a Theorem. See the Cozy documentation for an explanation of how to use apply. Cozy can only apply DivideEqn to an equation that looks exactly like c(...) = c(...). For example, it cannot be applied to the equation c = ca+cb. Instead, you would first rewrite it as c.1 = c(a+b) using the algebra rule and then apply DivideEqn. In particular, note that cab means (ca)b in Cozy because multiplication associates to the left, so you would need to explicitly transform cab = cde to c(ab) = c(de) using algeaba before you can divide both sides by c. ■ Remember that Cozy expects a "*" for multiplication. It will misunderstand ca = cb. You have to write that as c⭑a = c*b in Cozy, even though we would write it as ca = cb for a human reader. Submit and check your formal proof here: http://cozy.cs.washington.edu You can make as many attempts as needed to find a correct answer. (c) [3 Points] Translate your proof to English. This week, your English proofs will be graded not just for effort, but also for correctness. Keep in mind that your proof will be read by a human, so you need to explain the algebra steps in more detail than you would with Cozy. (Follow the same rules as in Problem 1.) Let domain of discourse be the integers, and let n and c be nonzero integers. Consider the following claim: Va Vb ((ca cn cb) → (a =n b)) This last "" is actually an "", but to make the problem easier, we have only asked you to prove "→". (a) [1 Point] Translate the claim into English. (b) [12 Points] Write a formal proof that the claim holds. Some important notes: Cozy has special notation for the predicate "=", but for everything else, it uses predicate notation. In particular, a b is written Divides (a, b), and a = b is written Congruent (a, b, m) ■ You will need to make use of Cozy's cite rule, which cite's a known theorem. In this case, the theorem you want to use is DivideEqn, which says the following about integers: " Va VbVc ((ca = cb) A(c =0)) + (a = b)) Cozy's apply rule is an even easier way to cite and use a Theorem. See the Cozy documentation for an explanation of how to use apply. Cozy can only apply DivideEqn to an equation that looks exactly like c(...) = c(...). For example, it cannot be applied to the equation c = ca+cb. Instead, you would first rewrite it as c.1 = c(a+b) using the algebra rule and then apply DivideEqn. In particular, note that cab means (ca)b in Cozy because multiplication associates to the left, so you would need to explicitly transform cab = cde to c(ab) = c(de) using algeaba before you can divide both sides by c. ■ Remember that Cozy expects a "*" for multiplication. It will misunderstand ca = cb. You have to write that as c⭑a = c*b in Cozy, even though we would write it as ca = cb for a human reader. Submit and check your formal proof here: http://cozy.cs.washington.edu You can make as many attempts as needed to find a correct answer. (c) [3 Points] Translate your proof to English. This week, your English proofs will be graded not just for effort, but also for correctness. Keep in mind that your proof will be read by a human, so you need to explain the algebra steps in more detail than you would with Cozy. (Follow the same rules as in Problem 1.) Let domain of discourse be the integers, and let n and c be nonzero integers. Consider the following claim: Va Vb ((ca cn cb) → (a =n b)) This last "" is actually an "", but to make the problem easier, we have only asked you to prove "→". (a) [1 Point] Translate the claim into English. (b) [12 Points] Write a formal proof that the claim holds. Some important notes: Cozy has special notation for the predicate "=", but for everything else, it uses predicate notation. In particular, a b is written Divides (a, b), and a = b is written Congruent (a, b, m) ■ You will need to make use of Cozy's cite rule, which cite's a known theorem. In this case, the theorem you want to use is DivideEqn, which says the following about integers: " Va VbVc ((ca = cb) A(c =0)) + (a = b)) Cozy's apply rule is an even easier way to cite and use a Theorem. See the Cozy documentation for an explanation of how to use apply. Cozy can only apply DivideEqn to an equation that looks exactly like c(...) = c(...). For example, it cannot be applied to the equation c = ca+cb. Instead, you would first rewrite it as c.1 = c(a+b) using the algebra rule and then apply DivideEqn. In particular, note that cab means (ca)b in Cozy because multiplication associates to the left, so you would need to explicitly transform cab = cde to c(ab) = c(de) using algeaba before you can divide both sides by c. ■ Remember that Cozy expects a "*" for multiplication. It will misunderstand ca = cb. You have to write that as c⭑a = c*b in Cozy, even though we would write it as ca = cb for a human reader. Submit and check your formal proof here: http://cozy.cs.washington.edu You can make as many attempts as needed to find a correct answer. (c) [3 Points] Translate your proof to English. This week, your English proofs will be graded not just for effort, but also for correctness. Keep in mind that your proof will be read by a human, so you need to explain the algebra steps in more detail than you would with Cozy. (Follow the same rules as in Problem 1.)
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Related Book For
Income Tax Fundamentals 2013
ISBN: 9781285586618
31st Edition
Authors: Gerald E. Whittenburg, Martha Altus Buller, Steven L Gill
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