Concept explainers
Assign a grade of A (correct), C (partially correct), or F (failure) to each. Justify assignments of grades other than A.
(a) Claim. Every polynomial of degree 3 with real coefficients has a real zero.
“Proof.” The polynomial
(b) Claim. There is a unique polynomial whose first derivative is
“Proof.” The antiderivative of
(c) Claim. Every prime number greater than 2 is odd.
“Proof.” The prime numbers greater than 2 are
(d) Claim. There exists an irrational number r such that
“Proof.” If
(e) Claim. For every real number x,
“Proof.” We proceed by three cases:
Case 1.
Case 2.
Case 3.
(f ) Claim. If x is prime, then
“Proof.” Let x be a prime number. If
(g) Claim. If t is an irrational number, then
“Proof.” Suppose there exists an irrational number t such that
(h) Claim. For real numbers x and y, if
“Proof.”
Case 1. If
Case 2. If
In either case,
(i) Claim. For every real number x in the interval
“Proof.” Assume that x is in the interval
( j) Claim. For every natural number n,
“Proof.” Let n be a natural number. Since n is a natural number,
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A Transition to Advanced Mathematics
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