Are mutually exclusive events also independent?

Cause in the case of a coin toss, the probability of getting heads is 0.5. But If i tossed it once and I get heads, it doesn't affect the probability when I toss it the second time. That's independent event?

However, wouldn't that also mean that since every toss that I do, whatever the result I get, does not affect subsequent tosses and for the same tosses, when i get heads, tails cannot occur and vice versa. Wouldn't that mean that the event are mutually exclusive AND independent?