C310 PLT DEPARTING BDR HAS AN NMAC WITH AN AA5 ENTERING THE TFC PATTERN.

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WE WERE DEPARTING RWY 6 ON A FULL-LENGTH DEP. RWY 11 WAS ALSO IN USE. DURING THE BACK TAXI; TWR SPOKE TO ANOTHER ACFT; BUT I DIDN'T REALLY PAY TOO MUCH ATTN TO IT; BECAUSE IT WAS NOT FOR OUR ACFT; AND I WAS CONFIRMING TKOF CHKLIST ITEMS. DURING CLBOUT AT ABOUT 1000 FT MSL/AGL; JUST BEFORE I WAS GOING TO BEGIN MY ON COURSE TURN NBOUND; THE TCASII SYS WENT OFF YELLING 'TFC; TFC.' I QUICKLY GLANCED AT THE SCREEN; AND NOTED AN INTRUDER ACFT APPROX 1/4-1/2 MI AWAY AT MY 1230 O'CLOCK POS AND 300 FT FROM MY ALT; AND DSNDING. I QUICKLY ACQUIRED THE ACFT VISUALLY; AND TOOK EVASIVE ACTION NBOUND. I CALLED THE LCL CTLR; AND ASKED IF HE HAD ANY TFC TO THE E OF THE FIELD. HE SAID 'YES; A GRUMMAN; BUT HE SHOULD NOT BE A FACTOR.' I SAID 'WELL; HE WAS A FACTOR; BECAUSE HE JUST BLEW PAST OUR R WING; AND THE TCASII WENT OFF!' THE GRUMMAN RESPONDED THAT 'IT WASN'T REALLY THAT CLOSE;' BUT I FOUND OUT LATER WHY IT DIDN'T SEEM AS CLOSE TO HIM; BECAUSE THE XMISSION I HEARD WHILE ON MY BACK TAXI; BUT DIDN'T LISTEN TO; WAS THE TWR TELLING THE GRUMMAN THAT I WAS DEPARTING RWY 6; TO LOOK OUT FOR ME. IF THE TWR SAID TO ME HE DIDN'T THINK THE GRUMMAN WOULD BE A FACTOR; WHY DID HE ISSUE US AS TFC TO THE GRUMMAN? IN REFLECTION; THE SIT WOULD HAVE BEEN A NON-EVENT IF THE LCL CTLR HAD TOLD ME ABOUT THE INBOUND GRUMMAN; SO I WOULD HAVE BEGUN MY ON COURSE TURN MUCH SOONER.