Condition for Difference of Field Elements to be Zero

Theorem

Let $\struct {F, +, \times}$ be a field whose zero is $0_F$.

Let $a, b \in F$

Then:

$a - b = 0_F$

if and only if

$a = b$

where $a - b$ denotes subtraction.

Proof

Necessary Condition

Let $a = b$.

Then:

\(\ds a - b\)

\(=\)

\(\ds a + \paren {-b}\)

Definition of Field Subtraction

\(\ds \)

\(=\)

\(\ds b + \paren {-b}\)

as $a = b$

\(\ds \)

\(=\)

\(\ds 0_F\)

Field Axiom $\text A4$: Inverses for Addition

$\Box$

Sufficient Condition

Let $a - b = 0_F$.

Then:

\(\ds a + \paren {-b}\)

\(=\)

\(\ds 0_F\)

Definition of Field Subtraction

\(\ds \leadsto \ \ \)

\(\ds \paren {a + \paren {-b} } + b\)

\(=\)

\(\ds 0_F + b\)

adding $b$ to both sides

\(\ds \leadsto \ \ \)

\(\ds a + \paren {\paren {-b} + b}\)

\(=\)

\(\ds 0_F + b\)

Field Axiom $\text A1$: Associativity of Addition

\(\ds \leadsto \ \ \)

\(\ds a + 0_F\)

\(=\)

\(\ds 0_F + b\)

Field Axiom $\text A4$: Inverses for Addition

\(\ds \leadsto \ \ \)

\(\ds a\)

\(=\)

\(\ds b\)

Field Axiom $\text A3$: Identity for Addition

$\blacksquare$

Sources

• 1973: C.R.J. Clapham: Introduction to Mathematical Analysis ... (previous) ... (next): Chapter $1$: Axioms for the Real Numbers: $2$. Fields: Theorem $2 \ \text {(ii)}$