How to Guess What to Prove Example
... [The result is true, but the following proof is not:] Proof: By strong induction. Let P (n) be the statement that n has a unique factorization. We prove P (n) for n > 1. Basis: P (2) is clearly true. Induction step: Assume P (2), . . . , P (n). We prove P (n + 1). If n + 1 is prime, we are done. If ...
... [The result is true, but the following proof is not:] Proof: By strong induction. Let P (n) be the statement that n has a unique factorization. We prove P (n) for n > 1. Basis: P (2) is clearly true. Induction step: Assume P (2), . . . , P (n). We prove P (n + 1). If n + 1 is prime, we are done. If ...
MTH4110/MTH4210 Mathematical Structures
... that not every input has an output, and, in particular, this cannot be the reason why f is not surjective. (b) At the heart of this is a common confusion, namely mixing up input and output of a function. Using the definition of f , the assertion “ f (1) has no input” is the same as “2 has no input” ...
... that not every input has an output, and, in particular, this cannot be the reason why f is not surjective. (b) At the heart of this is a common confusion, namely mixing up input and output of a function. Using the definition of f , the assertion “ f (1) has no input” is the same as “2 has no input” ...
Expectation and Moment Generating Function
... In previous chapter, we have studied the concepts of random or non-deterministic experiments followed by the classical definition of Probability function and the axiomatic approach to the Probability Theory. One-variate distribution theory in which discrete and continuous random variables and the di ...
... In previous chapter, we have studied the concepts of random or non-deterministic experiments followed by the classical definition of Probability function and the axiomatic approach to the Probability Theory. One-variate distribution theory in which discrete and continuous random variables and the di ...
Fundamental theorem of calculus
The fundamental theorem of calculus is a theorem that links the concept of the derivative of a function with the concept of the function's integral.The first part of the theorem, sometimes called the first fundamental theorem of calculus, is that the definite integration of a function is related to its antiderivative, and can be reversed by differentiation. This part of the theorem is also important because it guarantees the existence of antiderivatives for continuous functions.The second part of the theorem, sometimes called the second fundamental theorem of calculus, is that the definite integral of a function can be computed by using any one of its infinitely-many antiderivatives. This part of the theorem has key practical applications because it markedly simplifies the computation of definite integrals.