Solutions to Problem Set 2

Math 345/504

1-29-2018

1. An operation $\ast$ is defined on the set $\integer$ of integers by

$$a \ast b = a + b - 5.$$

(a) Show that $\ast$ is associative and commutative.

(b) Find the identity element for $\ast$ , verifying that the identity axiom holds.

(c) Find a formula for the inverse of an element x, verifying that the inverse axiom holds.

(a) Let $a, b, c \in \integer$ .

$$a \ast (b \ast c) = a \ast (b + c - 5) = a + (b + c - 5) - 5 = a + b + c - 10,$$

$$(a \ast b) \ast c = (a + b - 5) \ast c = (a + b - 5) + c - 5 = a + b + c - 10.$$

Therefore, $\ast$ is associative.

$$a \ast b = a + b - 5 = b + a - 5 = b \ast a.$$

Therefore, $\ast$ is commutative.

(b)

$$5 \ast a = 5 + a - 5 = a \quad\hbox{and}\quad a \ast 5 = a + 5 - 5 = a.$$

Therefore, 5 is the identity for $\ast$ .

(c)

$$(10 - a) \ast a = (10 - a) + a - 5 = 5 \quad\hbox{and}\quad a \ast (10 - a) = a + (10 - a) - 5 = 5.$$

Therefore, $10 - a$ is the inverse of a under $\ast$ .


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