Solutions To Mathematics Textbooks/Proofs and Fundamentals/Chapter 2
If is a real number, then the area of a circle of radius is .
If there is a line and a point not on , then there is exactly one line containing that is parallel to .
If is a triangle with sides of length and then
If is a continuous function on [a, b] and is an function such that , then...
If , then there is an integer q such that . Let q = n.
If , then there is an integer q such that . Let q = 1.
If , then there is an integer q such that . This implies , and so , and thus .
If n is an even integer, then for some integer k, .
If n is an odd integer, then for some integer k, .
If n is even, then . For integers j and k, let .
, so is even.
If n is odd, then . For integers j and k, let .
, so is odd.
If a|b, and b|bm then a|bm, implying aj = bm for some integer j.
Also, if a|c, and c|cn then a|cn, implying ai = cn for some integer i.
We let x = (j+i).
ax = aj+ai
ax = bm+cn
Which implies a|(bm+cn).
implies that for some integer, x.
implies that for some integer, y.
for some integer, j.
Let , hence .
Suppose that . This means there is an integer such that . Then, we have:
We may consider the integer . Therefore, we have that . Then