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I’m not a native English speaker and not a good math reader too. My question came while I reading this Debreu article about a existence of a real function to represent the preferences and I’m stuck in this passage:

If $A$, $B$ are two $f$-sets (resp. $i$-sets) and $A \cap B$ is not empty, then $A \cup B$ is an $f$-set (resp. $i$-set).

What’s this “resp.” means?

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It abbreviates the word respectively. In this context it means that two results are being stated simultaneously:

  • if $A,B$ are two $f$-sets, and $A\cap B\ne\varnothing$, then $A\cup B$ is an $f$-set; and
  • if $A,B$ are two $i$-sets, and $A\cap B\ne\varnothing$, then $A\cup B$ is an $i$-set.

In other words, read it first without the parenthetical items, and then read it again with the parenthetical items substituted for the items immediately before them.

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Resp. is an abbreviation for respectively.

What was written there is a shorter way of saying the following.

If $A,B$ are two f-sets and $A\cap B$ is not empty, then $A\cup B$ is an f-set.

If $A,B$ are two i-sets and $A\cap B$ is not empty, then $A\cup B$ is an i-set.

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