Solution

I open the lid of the basket which says "Eggs and Tomatoes" and see what it contains. Since the basket has been incorrectly labeled as "Eggs and Tomatoes" it does not contain both eggs and tomatoes; it either contains only eggs or only tomatoes. So, if I draw out an egg from the basket, it means the basket contains only eggs. On the other hand, if I happen to draw out a tomato, it means the basket contains only tomatoes.

Let's suppose that I draw out an egg from the basket labelled "Eggs and Tomatoes". This implies it contains only eggs and I must label it "Eggs" after removing the wrong "Eggs and Tomatoes" label. I, therefore, remove the "Eggs and Tomatoes" label and keep it aside. I, then, remove the label from the basket which has been incorrectly labelled "Eggs" and attach it to the basket which was incorrectly labelled "Eggs and Tomatoes".

So, now I have a basket which has been correctly labelled ("Eggs"), an unlabelled basket, a basket incorrectly labelled as "Tomatoes", and the label "Eggs and Tomatoes" which I have kept aside.

Let me push away the eggs basket now that it is correctly labelled. The other two baskets have to be labelled correctly without peeking into either of them. The basket labelled "Tomatoes" does not contain tomatoes (because it is incorrectly labelled). The label, therefore, has to come off and the only basket it can be attached to is the present unlabelled one (which earlier was labelled "Eggs"). That leaves the "Eggs and Tomatoes" label which had been kept aside to be attached to the basket that was earlier labelled as "Tomatoes". So, now the correct labels are in place, and we could correctly label the baskets by looking only into the "Eggs and Tomatoes" basket.