Looking for duplicates in columns between 2 dataframes in pandas Python

Question

How would I be able to write a function that detects if there are duplicates of a pandas Dataframe. So if I compare the index column between first and second there are no duplicates. But if I compare the index column between first and third there are duplicates of 1. I want to write a function that returns a bool of True when there are duplicates and a False when there aren't.

import pandas as pd

first = pd.DataFrame({'index': [1,4,5,6],
                      'vals':[3,4,5,7] })

second = pd.DataFrame({'index': [13,7,8,9],
                      'vals':[3,2,3,1] })

third = pd.DataFrame({'index': [1,11,2,12],
                      'vals':[6,7,51,2] })

Expected Output:

first and second: False
first and third: True