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A cognition scheme, whereby nodes are enabled with the full-duplex technology at the PHY layer in conjunction with packet fragmentation at the MAC layer, is proposed to increase the performance of cognitive radio networks. As full-duplex allows doubling the throughput by equipping secondary users (SUs) with the capability to sense and transmit simultaneously, it also enables them to inflict considerably less interference to primary users (PUs), and to discover spectrum holes for transmission opportunities more efficiently. On the other hand, fragmentation technique at the MAC layer provides better quality-of-service for the SUs by empowering the information forwarding in the unpredictable PUs' channels in a progressive manner. Selecting the effective data transfer, the successful packet transmission time and the energy efficiency as metrics, we assess the performance of this cognition scheme and demonstrate its overall merits in cognitive radio networks, in comparison with conventional schemes that use half-duplex and no packet fragmentation.