2.1 Binary Operations

1.
a. This is not commutative: 1 ∗ 2 = − 1 while 2 ∗ 1 = 1. It is not associative: (2 ∗ 1) ∗ 3 = 1 ∗ 3 = − 2, while 2 ∗ (1 ∗ 3) = 2 ∗ (− 2) = 4. There is no unity: If ea = a for all a, then ea = a so e = 2a for all a. This is impossible.
c. This is commutative: ab = a + bab = b + aba = ba. It is associative:

img

and, similarly, this equals (ab) ∗ c. The unity is 0:

img

Every a ≠ 1 has an inverse

img

e. This is not commutative (p, q) ∗ (p′, q′) = (p, q′) while (p′, q′) ∗ (p, q) = (p′, q) . It is associative:

img

There is no unity: If (a, b) ∗ (p, q) = (p, q) for all p, q, then a = p for all p.
g. This is commutative: gcd (n, m) = gcd (m, n). It is associative: Write d = gcd (k, m), d′ = gcd (m, n). Then d1 = (km) ∗ n = gcd (d, n), so d1 img d and d1 img n. But then, d1 k, d1 m and d1 n. It follows that d1 k and d1 d′, so d1 gcd (k, d′) = k ∗ (mn

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