Example
The partial map f from Z × Z onto Q which maps (x, y) to x/y provides an interpretation of the field Q of rational numbers in the ring Z of integers (to be precise, the interpretation is (2, f)). In fact, this particular interpretation is often used to define the rational numbers. To see that it is an interpretation (without parameters), one needs to check the following preimages of definable sets in Q:
- the preimage of Q is defined by the formula φ(x, y) given by ¬ (y = 0);
- the preimage of the diagonal of Q is defined by the formula φ(x1, y1, x2, y2) given by x1 × y2 = x2 × y1;
- the preimages of 0 and 1 are defined by the formulas φ(x, y) given by x = 0 and x = y;
- the preimage of the graph of addition is defined by the formula φ(x1, y1, x2, y2, x3, y3) given by x1×y2×y3 + x2×y1×y3 = x3×y1×y2;
- the preimage of the graph of multiplication is defined by the formula φ(x1, y1, x2, y2, x3, y3) given by x1×x2×y3 = x3×y1×y2.
Read more about this topic: Interpretation (model Theory)
Famous quotes containing the word example:
“Our intellect is not the most subtle, the most powerful, the most appropriate, instrument for revealing the truth. It is life that, little by little, example by example, permits us to see that what is most important to our heart, or to our mind, is learned not by reasoning but through other agencies. Then it is that the intellect, observing their superiority, abdicates its control to them upon reasoned grounds and agrees to become their collaborator and lackey.”
—Marcel Proust (1871–1922)