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1>in case of u(-t),the independent variable is reversed but in case of -u(t),the function itself reversed.
2> unit step function is not an odd function.so we can not write u(-t)=-u(t)
3>if u plot these functions independtenly then u will see that u(-t) lies in second quadrant and -u(t) lies in fourth quadrant.
This relation is not correct.
The proof is very easy:
write u(t)=1 as t>0 ,0 as t<0
put -t instead of t you get:
u(-t)=1 as -t>0 i.e. t<0 and 0 as -t<0 i.e. t>0
So we get that:
u(-t)= 1 as t<0 and 0 as t>0
while -u(t)= -1 as t>0,and 0 as t<0
so it's evident that u(-t) is not equal to -u(t)
Consider the four quadrants that we study with reference to trignometry.
For y(t) = u(t) ---> sketch is in the 1st quadrant
y(t) = -u(t) ----> sketch is in the 4th quadrant i.e. reflection of u(t) about the x-axis
y(t) = u(-t) ----> sketch is in the 2nd quadrant
y(t) = -u(-t) ----> sketch is in the 3rd quadrant.
they are not equal because u(-t) is on the left side and has a value 1 throughout but -u(t) is on the right side and a value of -1 throughout
if you mean to say in non signal terms then it is equal only for odd functions