What happens to your Social Security Number when you shed US citizenship or LPR status ?‏

Interestingly, there appears to be no statutory or regulatory rule in the law that allows an individual to somehow “expunge” or otherwise terminate their SSN, once obtained, even after loss of USC or LPR status.

"Relinquishing” citizenship means to give up U.S. citizenship voluntarily by committing any of the expatriating acts described in INA § 349(a), 8 U.S.C. § 1481(a).  One of the expatriating acts described in INA § 349(a)/8 U.S.C. § 1481(a) is “renouncing” or as it says in the statute, “… (5) making a formal renunciation of nationality before a diplomatic or consular officer of the United States in a foreign state, in such form as may be prescribed by the Secretary of State.”

Renouncing citizenship is a way to relinquish it, so when discussing this form of relinquishment, the two words can be used interchangeably.
For immigration purposes, the distinction between renunciation and the other forms of relinquishment may become meaningful when a former citizen is appealing or challenging the loss of nationality because establishing a lack of intent or duress during a formal renunciation is much more difficult than establishing a lack of intent when performing one of the other expatriating acts.  See 7 FAM 1211 (h).