Question: 1 consider the following statement ...
1. Consider the following statement: ∃𝑥, 𝑦, 𝑧 ∈ 𝑍 + | 𝑥 2 + 𝑦 2 = 𝑧 2 Which of the following are equivalent ways of expressing this statement? Justify your answer.
a. An integer squared added to an integer squared will remain the square of an integer.
b. The sum of two squared integers will, on occasion, remain a squared integer.
c. The sum of two integers, each a square of a positive integer, can be the square of a positive integer.
d. There exist at least three positive integers such that the square of the first plus the square of the second equals the square of the third.
e. Any integer squared can be expressed as the sum of two squared integers.
f. 3 2 + 4 2 = 5 2
g. There exist Pythagorean triples.
[Look up the definition of Pythagorean triple.]
Reading the statement left-to-right, we get, for example, “There exist at least three positive integers (which may or may not be distinct) such that the square of the first added to the square of the second equals the square of the third.” Notice that this is an existential statement; not a universal statement.