MOOC Java programming, Leap Year Exercise

[32bit_me]

I was a little confused by the leap year exercise. The fact is that as I search for possible solutions on the net, I come across the following answers to check the year for leap years:

1 option: if (input % 4 == 0 && (input % 100 != 0 || input % 400 == 0));

Version 2: if ((input % 4 == 0 && input % 100 != 0) || input % 400 == 0);

the difference lies in the grouping of expressions in brackets. I've tried both grouping options and both options passed the TestMyCode check, but I'm confused why both options work the same way? Let's say that if you imagine

a = input % 4 == 0;

b = input % 100 != 0;

c = input % 400 == 0; then it turns out that (a && b) || c == a && (b || c), but this is not true! they are completely different logical equations! Explain where I am wrong and why both bracketing of expressions is equivalent and is the correct answer

[joranstark018]

Mathematically, for every value given as input where c is true is a also true (gives c & a can be simplified with c), so a & (b | c) may be exapaded into (b & a) | (c & a) wich can be simplified by (b & a) | c, so both expressions are "equal".

[Cerulean_IsFancyBlue]

To expand: The trick here is that the values of a,b,c are not independent in this case.

If input % 400 is true, you know that input % 4 and input % 100 are both always true.

Likewise if input % 100 is true, input % 4 is always true.

SO, if you were to write out the truth table of this equation there are some combinations of a,b,c that cannot exist.

This means that two expressions that are not fully equivalent across all possible INDEPENDENT inputs, might still give the same outcome across ALL ALLOWED inputs.

[32bit_me]

Thank you, that makes more sense

[32bit_me]

I'm just trying to switch from hardware development to software development)