Intersection of Congruence Classes/Examples/Intersection of 1 mod 3 with -1 mod 4
Jump to navigation
Jump to search
Examples of Use of Intersection of Congruence Classes
- $\eqclass 1 3 \cap \eqclass {-1} 4 = \eqclass 7 {12}$
Proof
From Intersection of Congruence Classes:
- $\eqclass 1 3 \cap \eqclass {-1} 4 = \eqclass x {12}$
for some $x$ which is to be found.
We have that:
\(\ds \eqclass 1 3\) | \(=\) | \(\ds \eqclass 1 {12} \cup \eqclass {3 + 1} {12} \cup \eqclass {2 \times 3 + 1} {12} \cup \eqclass {3 \times 3 + 1} {12}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \eqclass 1 {12} \cup \eqclass 4 {12} \cup \eqclass 7 {12} \cup \eqclass {10} {12}\) |
and that:
\(\ds \eqclass {-1} 4\) | \(=\) | \(\ds \eqclass {4 + \paren {-1} } {12} + \eqclass {2 \times 4 + \paren {-1} } {12} + \eqclass {3 \times 4 + \paren {-1} } {12}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \eqclass 3 {12} \cup \eqclass 7 {12} \cup \eqclass {11} {12}\) |
Hence the result by taking the intersection.
$\blacksquare$
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $2$. Equivalence Relations: Exercise $8$