Definition, Proposition and Theorem：練習問題

A statement that is accepted as true without the need for a proof is called a(n):

Consider the following logical structure in a textbook:
1. An integer \( n \) is perfect if the sum of its proper divisors equals \( n \).
2. If \( 2^p - 1 \) is prime, then \( 2^{p-1}(2^p - 1) \) is a perfect number.
Which terms correctly describe statements 1 and 2 respectively?

In the context of mathematical logic, which of the following is the best description of a proposition?

Which of the following is the negation of the statement: "For every real number \(x\), there exists a real number \(y\) such that \(x + y > 0\)"?

Which term is used to describe a proposition that follows as a direct and easy consequence of a theorem that has already been proven?

In mathematical logic, explain why the statement "Please solve this equation" is not considered a proposition.

Let \(n\) be an integer. Let \(P\) be the statement "\(n\) is a multiple of 12" and \(Q\) be the statement "\(n\) is a multiple of 3". Determine if \(P\) is a necessary or sufficient condition for \(Q\), and justify your answer.

State the converse of the following proposition: "If a function \(f\) is differentiable at \(x = a\), then \(f\) is continuous at \(x = a\)."

In geometry, consider the following Definitions for a quadrilateral \( ABCD \):
1. A rectangle is a quadrilateral where all four interior angles \( \angle A, \angle B, \angle C, \angle D \) are \( 90^\circ \).
2. A parallelogram is a quadrilateral where opposite sides are parallel.
(a) Based on the Theorem "Every rectangle is a parallelogram," is having four right angles a sufficient condition for a quadrilateral to be a parallelogram? Justify your answer.
(b) Is being a parallelogram a necessary condition for a quadrilateral to be a rectangle? Explain.

A Definition states that a prime number is a natural number greater than 1 that is divisible only by 1 and itself. Consider the Proposition: "If \( n \) is a prime number and \( n > 2 \), then \( n \) is odd."
(a) Identify the logical structure of this proposition and state its contrapositive.
(b) Prove the proposition by showing that its contrapositive is true (i.e., if \( n \) is an even integer greater than 2, then \( n \) is not prime).
(c) State the inverse of the proposition and provide a counterexample to show that the inverse is false.