Intersection of Sets of Integer Multiples/Examples/(3 Z cap 6 Z) cup 18 Z
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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 $2$