Prove the following “laws of algebra” for ℝ, using only axioms (1)–(5):
(a) If x + y = x, then y = 0.
(b) 0 • x = 0. [Hint: Compute (x + 0) • x.]
(c) −0 = 0.
(d) −(−x) = x.
(e) x(−y) = −(xy) = (−x)y.
(f) (−1)x = −x.
(g) x(y − z) = xy − xz.
(h) −(x + y) = −x − y; −(x − y) = −x + y.
(i) If x ≠ 0 and x • y = x, then y = 1.
(j) x/x = 1 if x ≠ 0.
(k) x/1 = x.
(l) .
(m) (1/y)(1/z) = 1/(yz) if y, z ≠ 0.
(n) (x/y)(w/z) = (xw)/(yz) if y, z # 0.
(o) (x/y) +- (w/z) = (xz + wy)/(yz) if y, z ≠ 0.
(p)
(q) 1 /(w/z) = z/w if w z ≠ 0.
(r) (x/y)/(w/z) = (xz)/(yw) if y, w, z ≠ 0.
(s) (ax)/y = a(x/y) if y ≠ 0.
(t) (−x)/y = x/(−y) = −(x/y) if y ≠ 0.
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