Intersection of Sets of Integer Multiples/Examples/(3 Z cap 6 Z) cup 18 Z

Examples of Use of Intersection of Sets of Integer Multiples

Let $m \Z$ denote the set of integer multiples of $m$.

Then:

$\paren {3 \Z \cap 6 \Z} \cup 18 \Z = 6 \Z$

Proof

From Intersection of Sets of Integer Multiples:

$3 \Z \cap 6 \Z = \lcm \set {3, 6} \Z = 6 \Z$

Then we have that:

$18 \Z \subseteq 6 \Z$

and so from Union with Superset is Superset:

$6 \Z \cup 18 \Z = 6 \Z$

Hence the result.

$\blacksquare$

Sources

• 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 1$. Sets; inclusion; intersection; union; complementation; number systems: Exercise $2$