The number of people employed in May declined for first time in 6 months.

[Statistical]

Unemployment only went down because people "gave up".

"Unemployment" (U3) doesn't include all the people with "no job". U3 doesn't have anything to do with unemployment insurance either (another urban legend). U3 is specifically the number of people in workforce population (working age, not retired, able to work, not in prison, not disabled, etc) who are currently willing & able to work, actively looking for employment and not currently employed. So a lot of people are missed in U3 number (about 2.4 million Americans).

A simpler (less easy to manipulate number) is to look at how many people are working in the US.

ftp://ftp.bls.gov/pub/suppl/empsit.cpseea1.txt

Number of persons employed:
Dec - 137,792,000
Jan - 138,333,000 (gain 541,000)
Feb - 138,641,000 (gain 308,000)
Mar - 138,905,000 (gain 264,000)
Apr - 139,455,000 (gain 555,000)
May - 139,420,000 (minus 35,000)

In May less Americans are working than in April. Period. No way to manipulate or spin that number.

The only reason unemployment went "down" is because unemployment (U3) requires you to be recently looking for a job. More people stopped looking thus unemployment "went down". It would be like BP saying number of animals suffering from oil went down and it turns out the reason why is that more of them are now dead (and thus no longer suffering). :(

The raw data:
ftp://ftp.bls.gov/pub/suppl/empsit.cpseea1.txt

More here:
http://www.bls.gov/cps/tables.htm#monthly