Express the following using language of Predicate Calculus, whereit is understood that the people being discussed are in the courtroomIf any sentence is ambiguous, give all symbolic versions.(i)All judges are sober(ii)There is a dishonest lawyer.(iii)All defendants are innocent.(iv)Some plaintiffs are lawyers(v)Anybody who is honest and a defendant is innocent(vi)All defendants who are not sober are dishonest. 3. а)Prove by induction that11P(n):2+6+12+20+...+n(2n+2) = n(n+1)(n+2) Vn2123Let f :R→(-1,1) be defined by f(x) =,XeR.Find theX-1b)inverse of the above function if it exists, where R is the set ofreal numbers?

For part a, principle of mathematical induction is used. For part b, function is defined when imagee…

3. а) Prove by induction that 1 P(n):2+6+12+20+...+n(2n+2) =n(n+1)(n+ 2) Vn21 1 3 Let f:R→(-1,1) be defined by f(x) = X ,XeR. Find the X-1 b) inverse of the above function if it exists, where R is the set of real numbers?

3. а) Prove by induction that 1 P(n):2+6+12+20+...+n(2n+2) =n(n+1)(n+ 2) Vn21 1 3 Let f:R→(-1,1) be defined by f(x) = X ,XeR. Find the X-1 b) inverse of the above function if it exists, where R is the set of real numbers?

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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A(1, 1) The following function was defined by Ackermann. A(0, n) = n + 1 for all nonnegative intergers n A(m, 0) = A(m - 1,1) for all positive integers m A(m, n) = A(m - 1, A(m, n - 1)) for all positive integers m and n. Find A(1, 1). write down all intermediate steps to justify final answer.
answer with example please
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7. Consider the function f: N → N given by f(n) (a) Is f injective? Prove your answer. (b) Is f surjective? Prove your answer. ( n/2, if n is even (3n + 1, if n is odd
Let n be a positive integer and let f : [0..n] → [0..n] be an injective function. Define the function g : [0..n] → Z as g(x) = n - (f(x))². Prove that is also injective.
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Consider the following inductive definition of a version of Ackermann's function:
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