1) Which of the following sentences are statements? Determine, if possible, whet

Question

1) Which of the following sentences are statements? Determine, if possible, whether the statement is conditional or not and the truth value of each statement, but DO NOT attempt any proofs.

a) There exist nonzero integers a, b, and c with a3 + b3 = c3.

b) If p is a prime number, then p+2 is a prime number.

c) For infinitely many prime numbers p, p + 2 is also a prime number.

d) Let M be a manifold, and let : M S3 be a homeomorphism, where S3 is the three sphere.

e) If L is a linear operator between finite-dimensional vector spaces, then L admits a nontrivial invariant subspace.

Explanation / Answer

(According to Chegg policy only four subquestions will be answered. Please post the remaining in another question)

a) There exist nonzero integers a, b, and c with a3 + b3 = c3

This is a statement but not a conditional one. The truth value is false.

b) If p is a prime number, then p+2 is a prime number

This is a statement and is conditional. The truth value is false.

c) For infinitely many prime numbers p, p + 2 is also a prime number.

This is a statement and is not conditional. The truth value is true.

d) Let M be a manifold, and let : M S3 be a homeomorphism, where S3 is the three sphere.

This is not a statement. It has no truth value.

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