Exercise 4.1.4

Exercise 4.1.4 Prove the remaining identities in Proposition 4.1.6. (Hint: one can save some work by using some identities to prove others. For instance, once you know that \(xy = yx\), you get for free that \(x1 = 1x\), and once you also prove \(x(y + z) = xy + xz\), you automatically get \((y + z)x = yx + zx\) for free.)

• \(\bullet\) Show that \(x + y = y + x\).

• \(\Vdash\) \(x + y\)

• \(=\) { \(x = a — b\), \(y = c — d\) }

  • \((a — b) + (c — d)\)

• \(=\) { Definition 4.1.2 }

  • \((a + c) — (b + d)\)

• \(=\) { Commutativity of the addition on \(\mathbb N\) }

  • \((c + a) — (d + b)\)

• \(=\) { Definition 4.1.2 }

  • \((c — d) + (a — b)\)

• \(=\) { \(x = a — b\), \(y = c — d\) }

  • \(y + x\)

• \(\square\)

• \(\bullet\) Show that \((x + y) + z = x + (y + z)\).

• \(\Vdash\) \((x + y) + z\)

• \(=\) { \(x = a — b\), \(y = c — d\), \(z = e — f\) }

  • \(((a — b) + (c — d)) + (e — f)\)

• \(=\) { Definition 4.1.2 }

  • \(((a + c) — (b + d)) + (e — f)\)

• \(=\) { Definition 4.1.2 }

  • \(((a + c) + e) — ((b + d) + f)\)

• \(=\) { Associativity of the addition on \(\mathbb N\) }

  • \((a + (c + e)) — (b + (d + f))\)

• \(=\) { Definition 4.1.2 }

  • \((a — b) + ((c + e) — (d + f))\)

• \(=\) { Definition 4.1.2 }

  • \((a — b) + ((c — d) + (e — f))\)

• \(=\) { \(x = a — b\), \(y = c — d\), \(z = e — f\) }

  • \(x + (y + z)\)

• \(\square\)

• \(\bullet\) Show that \(x + 0 = 0 + x = x\).

• \(-\) Let \(c\in \mathbb N\) and \(0 := c — c\)

• \(\Vdash\) \(x + 0\)

• \(=\) { \(x = a — b\), \(0 = c — c\) }

  • \((a — b) + (c — c)\)

• \(=\) { Definition 4.1.2 }

  • \((a + c) — (b + c)\)

• \(=\) { Definition 4.1.1, Commutativity and associativity of the addition on \(\mathbb N\), \((a + c) + b = a + (b + c)\) }

  • \(a — b\)

• \(=\) { \(x = a — b\) }

  • \(x\)

• \(=\) { \(x = a — b\) }

  • \(a — b\)

• \(=\) { Definition 4.1.1, Commutativity and associativity of the addition on \(\mathbb N\), \(a + (c + b) = (c + a) + b\) }

  • \((c + a) — (c + b)\)

• \(=\) { Definition 4.1.2 }

  • \((c — c) + (a — b)\)

• \(=\) { \(x = a — b\), \(0 = c — c\) }

  • \(0 + x\)

• \(\square\)

• \(\bullet\) (Commutative law of \(\mathbb Z\)) Show that \(xy = yx\).

• \(\Vdash\) \(xy\)

• \(=\) { \(x = a — b\), \(y = c — d\) }

  • \((a — b)(c — d)\)

• \(=\) { Definition 4.1.2 }

  • \((ac + bd) — (ad + bc)\)

• \(=\) { Commutativity of the addition and the multiplication on \(\mathbb N\) }

  • \((ca + db) — (cb + da)\)

• \(=\) { Definition 4.1.2 }

  • \((c — d)(a — b)\)

• \(=\) { \(x = a — b\), \(y = c — d\) }

  • \(yx\)

• \(\square\)

• \(\bullet\) Show that \(x1_Z = 1_Zx = x\).

• \(-\) Let \(1_Z = 1_N — 0_N\), \(x = a — b\)

• \(\Vdash\) \(x1_Z\)

• \(=\) { \(1_Z = 1_N — 0_N\), \(x = a — b\) }

  • \((a — b)(1_N — 0_N)\)

• \(=\) { Definition 4.1.2 }

  • \((a1_N + b0_N) — (a0_N + b1_N)\)

• \(=\) { Multiplication on \(\mathbb N\) }

  • \(a — b\)

• \(=\) { \(x = a — b\) }

  • \(x\)

• \(=\) { \(x = a — b\) }

  • \(a — b\)

• \(=\) { Multiplication on \(\mathbb N\) }

  • \((1_Na + 0_Nb) — (1_Nb + 0_Na)\)

• \(=\) { Definition 4.1.2 }

  • \((1_N — 0_N)(a — b)\)

• \(=\) { \(1_Z = 1_N — 0_N\), \(x = a — b\) }

  • \(1_Zx\)

• \(\square\)

• \(\bullet\) (Left distributive law of \(\mathbb Z\)) Show that \(x(y + z) = xy + xz\).

• \(-\) Let \(x = a — b\), \(y = c — d\), \(z = e — f\)

• \(\Vdash\) \(x(y + z)\)

• \(=\) { \(x = a — b\), \(y = c — d\), \(z = e — f\) }

  • \((a — b)((c — d) + (e — f))\)

• \(=\) { Definition 4.1.2 }

  • \((a — b)((c + e) — (d + f))\)

• \(=\) { Definition 4.1.2 }

  • \((a(c + e) + b(d + f)) — (a(d + f) + b(c + e))\)

• \(=\) { Commutative and Distributive law of \(\mathbb N\) }

  • \(((ac + bd) + (ae + bf)) — ((ad + bc) + (af + be))\)

• \(=\) { Definition 4.1.2 }

  • \(((ac + bd) — (ad + bc)) + ((ae + bf) — (af + be))\)

• \(=\) { Definition 4.1.2 }

  • \((a — b)(c — d) + (a — b)(e — f)\)

• \(=\) { \(x = a — b\), \(y = c — d\), \(z = e — f\) }

  • \(xy + xz\)

• \(\square\)

• \(\bullet\) \((y + z)x = yx + zx\).

• \(\Vdash\) \((y + z)x\)

• \(=\) { Commutative law of \(\mathbb Z\) }

  • \(x(y + z)\)

• \(=\) { Left distributive law of \(\mathbb Z\) }

  • \(xy + xz\)

• \(=\) { Commutative law of \(\mathbb Z\) }

  • \(yx + zx\)

• \(\square\)