Intersection of Congruence Classes/Examples/Intersection of 1 mod 3 with -1 mod 4

Examples of Use of Intersection of Congruence Classes

$\eqclass 1 3 \cap \eqclass {-1} 4 = \eqclass 7 {12}$

$\eqclass 1 3 \cap \eqclass {-1} 4 = \eqclass x {12}$

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.

$\blacksquare$

1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $2$. Equivalence Relations: Exercise $8$