Let f a b and g : b -> c be maps.
(d) if g of is one-to-one and f is onto, show that g is one-to-one. (e) if go f is onto and g is one-to-one, show that f is onto.

Step-by-step explanation:

Let us denote probability of   spoilage as follows

Transformer spoilage = P( T ) ; line spoilage P ( L )

Both P ( T ∩ L ) .

Given

P( T )  = .05

P ( L ) = .08

P ( T ∩ L ) = .03

a )

For independent events

P ( T ∩ L ) =  P( T ) x  P ( L )

But  .03  ≠  .05 x .08

So they are not independent of each other .

b )

i )

Probability of line spoilage given that there is transformer spoilage

P L/ T ) = P ( T ∩ L ) / P( T )

= .03 / .05

= 3 / 5 .

ii )

Probability of transformer spoilage but not line spoilage.

P( T ) - P ( T ∩ L )

.05 - .03

= .02

iii )Probability of transformer spoilage given that there is no line spoilage

[ P( T ) - P ( T ∩ L ) ] / 1 - P ( L )

=  .02 / 1 - .08

= .02  / .92

= 1 / 49.

i v )

Neither transformer spoilage nor there is no line spoilage

= 1 - P ( T ∪ L )

1 - [  P( T ) +  P ( L ) - P ( T ∩ L ]

= 1 - ( .05 + .08 - .03 )

=  0 .9