We describe a new multi-frequency technique for breast cancer detection. Applying a constant voltage with multiple sinusoidal frequencies between a reference electrode on a distal part of a patient and a scan probe placed on the breast, we measure exit currents from an array of electrodes inside the probe that are kept at the ground potential. The distribution of measured exit currents is called the trans-admittance map and the instrument is called the trans-admittance scanner (TAS). We assume a three-dimensional homogeneous domain including an internal lesion with a complex conductivity different from that of the background. Mathematically analyzing the multi-frequency trans-admittance map obtained on the surface of the domain, we found that both conductivity and permittivity ratios between the background and the lesion are crucial in extracting any useful information about the lesion from the map. Choosing two frequencies in the range of 10 Hz to 500 kHz with one significantly lower than the other and assuming that conductivity values of the background and the lesion do not change much from the low to high frequency, the lesion underneath the probe can be detected only when the conductivity ratio between the background and the lesion is different from the permittivity ratio between the background and the lesion at the chosen high frequency. Results of numerical simulations and saline phantom experiments using a developed TAS system are well matched with the mathematical analysis. The biggest advantage of this multi-frequency technique is that we do not need separately measured reference data in the absence of any lesion. We suggest future studies of a more sophisticated lesion detection algorithm based on the analysis and findings described in this paper.