Variable-Binding Operation
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See: Mathematical Operation, Bound Variable, Free Variable.
References
2011
- http://en.wikipedia.org/wiki/Free_variables_and_bound_variables#Variable-binding_operators
- The following : [math]\displaystyle{ \sum_{x\in S} \quad\quad \prod_{x\in S} \quad\quad \int_0^\infty\cdots\,dx \quad\quad \lim_{x\to 0} \quad\quad \forall x \quad\quad \exists x \quad\quad \psi x \quad\quad }[/math] are variable-binding operators. Each of them binds the variable x.
Note that many of these are operators which act on functions of the bound variable. In more complicated contexts, such notations can become awkward and confusing. It can be useful to switch to notations which make the binding explicit, such as: [math]\displaystyle{ \sum_{1 \, \ldots \, 10} \left( k \mapsto f(k,n) \right) }[/math] for sums or [math]\displaystyle{ D \left( x \mapsto x^2 + 2x + 1 \right) \, }[/math] for differentiation.
- The following : [math]\displaystyle{ \sum_{x\in S} \quad\quad \prod_{x\in S} \quad\quad \int_0^\infty\cdots\,dx \quad\quad \lim_{x\to 0} \quad\quad \forall x \quad\quad \exists x \quad\quad \psi x \quad\quad }[/math] are variable-binding operators. Each of them binds the variable x.