What is the difference between equality and equivalence?

public final class CaseInsentiveitveString {
...
	// broken
	@Override public boolean equals(Object o) {
		if (o instance of CaseInsentiveitveString)
			return s.equalsIgnoreCase(
				((CaseInsentiveitveString) o).s);
		if (o instanceof String) // One-way interoperability!
			return s.equalsIgnoreCase((String) o);
		return false;
	}
...

}

null == 0   # true , null is equivelant to 0 ( in php )
null === 0  # false, null is not equal to 0 ( in php )

int i = 3;
double d = 3.0;

int i = 3;
double d = 3.0;

0.9999999999999999... = 1