Question

when constructing a direct proof for the theorem \for all integers n, if n is even, then n² is even,\ what is the initial assumption?
options:
assume n is an integer and n is odd.
assume n² is odd.
assume n is an integer and n² is even.
assume n is an integer and n is even.

(question 5) for the statement \if a quadrilateral is a square, then all its angles are right angles,\ the hypothesis is \a quadrilateral is a square.\
options: true, false

(question 6) in a direct proof of a conditional statement \if p, then c,\ what is assumed to be true at the beginning of the proof?
options:
the conclusion (c)
the hypothesis (p)
both the hypothesis (p) and the conclusion (c)
neither the hypothesis (p) nor the conclusion (c)

Explanation:

Question 1 (When constructing a direct proof...):

The theorem is "For all integers \( n \), if \( n \) is even, then \( n^2 \) is even". In a direct proof, we assume the hypothesis (the "if" part) to be true and then show the conclusion (the "then" part) follows. The hypothesis here is " \( n \) is an integer and \( n \) is even", so we assume that.

A conditional statement is "If \( P \), then \( Q \)", where \( P \) is the hypothesis and \( Q \) is the conclusion. The given statement is "If a quadrilateral is a square, then all its angles are right angles". So the hypothesis is "a quadrilateral is a square", which matches the given claim. Thus the statement is True.

In a direct proof of "If \( p \), then \( c \)", we start by assuming the hypothesis \( p \) is true and then derive the conclusion \( c \). So we assume the hypothesis (\( p \)) is true at the beginning.

Answer:

Assume \( n \) is an integer and \( n \) is even.

Question 5 (For the statement...):