# Definition:Agreement

## Definition

### Relations

Let:

$(1): \quad \mathcal R_1 \subseteq S_1 \times T_1$ be a relation on $S_1 \times T_1$
$(2): \quad \mathcal R_2 \subseteq S_2 \times T_2$ be a relation on $S_2 \times T_2$
$(3): \quad X \subseteq S_1 \cap S_2$

Let:

$\forall s \in X: \mathcal R_1 \left ({s}\right) = \mathcal R_2 \left ({s}\right)$

where $\mathcal R_1 \left ({s}\right)$ denotes the image of $s$ under $\mathcal R$:

$\mathcal R_1 \left ({s}\right) := \left\{ {t \in T: s \mathrel{\mathcal R_1} t}\right\}$

Then the relations $\mathcal R_1$ and $\mathcal R_2$ are said to agree on or be in agreement on $X$.

### Mappings

The concept is usually seen in the context of mappings:

Let:

$(1): \quad f_1: S_1 \to T_1$ be a mapping from $S_1$ to $T_1$
$(2): \quad f_2: S_2 \to T_2$ be a mapping from $S_2$ to $T_2$
$(3): \quad X \subseteq S_1 \cap S_2$

Let:

$\forall s \in X: \map {f_1} s = \map {f_2} s$

Then the mappings $f_1$ and $f_2$ are said to agree on or be in agreement on $X$.