Modulo Addition is Well-Defined

Theorem

Let $m \in \Z$ be an integer.

Let $\Z_m$ be the set of integers modulo $m$.

Let $\eqclass a m$ denote the equivalence class on $\Z_m$, for some $a \in \Z$.

The modulo addition operation on $\Z_m$, defined by the rule:

$\eqclass a m +_m \eqclass b m = \eqclass {a + b} m$

is a well-defined operation.

That is:

If $a \equiv b \pmod m$ and $x \equiv y \pmod m$, then $a + x \equiv b + y \pmod m$.

Corollary

It follows that:

$\eqclass a m -_m \eqclass b m = \eqclass {a - b} m$

is a well-defined operation.

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Let:

$a \equiv b \pmod z$

and:

$x \equiv y \pmod z$

where $a, b, x, y \in \R$.

Then:

$a + x \equiv b + y \pmod z$

Proof 1

We need to show that if:

$\eqclass {x'} m = \eqclass x m$

$\eqclass {y'} m = \eqclass y m$

then:

$\eqclass {x' + y'} m = \eqclass {x + y} m$

Since:

$\eqclass {x'} m = \eqclass x m$

and:

$\eqclass {y'} m = \eqclass y m$

it follows from the definition of set of integers modulo $m$ that:

$x \equiv x' \pmod m$

and:

$y \equiv y' \pmod m$

By definition, we have:

$x \equiv x' \pmod m \implies \exists k_1 \in \Z: x = x' + k_1 m$

$y \equiv y' \pmod m \implies \exists k_2 \in \Z: y = y' + k_2 m$

which gives us:

$x + y = x' + k_1 m + y' + k_2 m = x' + y' + \paren {k_1 + k_2} m$

As $k_1 + k_2$ is an integer, it follows that, by definition:

$x + y \equiv \paren {x' + y'} \pmod m$

Therefore, by the definition of integers modulo $m$:

$\eqclass {x' + y'} m = \eqclass {x + y} m$

$\blacksquare$

Proof 2

The equivalence class $\eqclass a m$ is defined as:

$\eqclass a m = \set {x \in \Z: x = a + k m: k \in \Z}$

That is, the set of all integers which differ from $a$ by an integer multiple of $m$.

Thus the notation for addition of two set of integers modulo $m$ is not usually:

$\eqclass a m +_m \eqclass b m$

What is more normally seen is:

$a + b \pmod m$

Using this notation:

$\ds a$

$\equiv$

$\ds b$

$\ds \pmod m$

$\, \ds \land \, $

$\ds c$

$\equiv$

$\ds d$

$\ds \pmod m$

$\ds \leadsto \ \ $

$\ds a \bmod m$

$=$

$\ds b \bmod m$

Definition of Congruence Modulo Integer

$\, \ds \land \, $

$\ds c \bmod m$

$=$

$\ds d \bmod m$

$\ds \leadsto \ \ $

$\ds a$

$=$

$\ds b + k_1 m$

for some $k_1 \in \Z$

$\, \ds \land \, $

$\ds c$

$=$

$\ds d + k_2 m$

for some $k_2 \in \Z$

$\ds \leadsto \ \ $

$\ds a + c$

$=$

$\ds b + d + \paren {k_1 + k_2} m$

Definition of Integer Addition

$\ds \leadsto \ \ $

$\ds a + c$

$\equiv$

$\ds b + d$

$\ds \pmod m$

Definition of Modulo Addition

$\blacksquare$

Examples

Modulo Addition: $2 + 3 \equiv -6 + 15 \pmod 4$

We have:

$\ds 2$

$\equiv$

$\ds -6$

$\ds \pmod 4$

Congruence Modulo $4$: $2 \equiv -6 \pmod 4$

$\ds 3$

$\equiv$

$\ds 15$

$\ds \pmod 4$

Congruence Modulo $4$: $3 \equiv 15 \pmod 4$

$\ds \leadsto \ \ $

$\ds 2 + 3 = 5$

$\equiv$

$\ds -6 + 15 = 9$

$\ds \pmod 4$

Modulo Addition: $19 + 6 \equiv 11 + 2 \pmod 4$

We have:

$\ds 19$

$\equiv$

$\ds 11$

$\ds \pmod 4$

$\ds 6$

$\equiv$

$\ds 2$

$\ds \pmod 4$

$\ds \leadsto \ \ $

$\ds 19 + 6 = 25$

$\equiv$

$\ds 11 + 2 = 13$

$\ds \pmod 4$

Sources

1964: Iain T. Adamson: Introduction to Field Theory ... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 1$. Rings and Fields: Example $4$
1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): Chapter $1$: Rings - Definitions and Examples: $2$: Some examples of rings: Ring Example $2$
1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $1$: Equivalence Relations: $\S 19 \beta$
1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 14.3 \ \text {(i)}$: Congruence modulo $m$ ($m \in \N$)
1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 2.3$: Congruences: Theorem $1 \ \text{(i)}$
1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.4$: Integer Functions and Elementary Number Theory: Law $\text{A}$
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): congruence (modulo $n$)
2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): congruence (modulo $n$)

Theorem

Corollary

Real Modulus

Proof 1

Proof 2

Examples

Modulo Addition: $2 + 3 \equiv -6 + 15 \pmod 4$

Modulo Addition: $19 + 6 \equiv 11 + 2 \pmod 4$

Sources