#### Mathematical English Usage - a Dictionary

##### by Jerzy Trzeciak

## define

[*see also*: declare, let, set

To define P_{n+1} from P_{n}, we let......

We define a <the> function f by <by setting> f=......

We define the set T to consist of those f for which......

We define a map to be simple if......

Now M is defined to be the set of all sums of the form......

Here F is only defined up to an additive constant.

Here u^{+} and u^{-} are the positive and negative parts of u, as defined in Section 5.

As defined in Section 1, these are structures of the form......

The function f so defined satisfies......

The notion of backward complete is defined analogously by
exchanging the roles of f and f^{-1}.

The fact that the number T(p) is uniquely defined, even though p is not, enables us to define the nullity of A as follows.

The other two defining properties of a σ-algebra are verified in the same manner.

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