# Intersection of a recognizable language and a decidable language is decidable?

I'm having trouble with proving that "Intersection of a recognizable language anda decidable language is decidable.

I assume this is true although I have no idea how to proof it. Can somebody point me in the right direction?

## migrated from cstheory.stackexchange.comJun 4 at 0:39

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• Looks like homework. Voting to close. – Aryeh Jun 2 at 9:53
• It's not homework, just extra questions in our course notes to think about to understand the subject matter better. – Bobby Jun 2 at 9:57

It's false. Let $$L_1=\Sigma^*$$ be a decidable language and $$L_2=L_{HALT}$$ be the (recognizable) language of all halting TM-string pairs. Then $$L_1\cap L_2=L_2$$, which is not decidable.