Is it correct to say:
My father's shoe
I want to point to the one shoe of that pair of my father's shoes.
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Sign up to join this communityYes. This is entirely valid. The two shoes together form a pair of shoes - one of which is a single shoe.
Hence in the sentence,
The dog bit my father's shoe
In this sentence, the dog bit one of my father's shoes. The sentence makes no attempt to indicate which of the shoes was bitten, but does make it clear by implication that only one of the two shoes was bitten.
If you want to be explicit, you can say which shoe by stating whether it was the left or right shoe that was bitten. (left and right are always from the perspective of if you were wearing the shoes)
The dog bit my father's left shoe.
The dog didn't bite my father's right shoe.
In this case the sentence makes it clear which of the two shoes was bitten (the left one).