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Departing from the classic paradigm of data-centric designs, the 6G networks for supporting edge AI features task-oriented techniques that focus on effective and efficient execution of AI task. Targeting end-to-end system performance, such techniques are sophisticated as they aim to seamlessly integrate sensing (data acquisition), communication (data transmission), and computation (data processing). Aligned with the paradigm shift, a task-oriented over-the-air computation (AirComp) scheme is proposed in this paper for multi-device split-inference system. In the considered system, local feature vectors, which are extracted from the real-time noisy sensory data on devices, are aggregated over-the-air by exploiting the waveform superposition in a multiuser channel. Then the aggregated features as received at a server are fed into an inference model with the result used for decision making or control of actuators. To design inference-oriented AirComp, the transmit precoders at edge devices and receive beamforming at edge server are jointly optimized to rein in the aggregation error and maximize the inference accuracy. The problem is made tractable by measuring the inference accuracy using a surrogate metric called discriminant gain, which measures the discernibility of two object classes in the application of object/event classification. It is discovered that the conventional AirComp beamforming design for minimizing the mean square error in generic AirComp with respect to the noiseless case may not lead to the optimal classification accuracy. The reason is due to the overlooking of the fact that feature dimensions have different sensitivity towards aggregation errors and are thus of different importance levels for classification. This issue is addressed in this work via a new task-oriented AirComp scheme designed by directly maximizing the derived discriminant gain.