Question

Try to prove the following using the general principle.

(i) (x − y) + z = x − (y − z) for all numbers x, y, z

(ii) (x + y) − z = x + (y − z) for all numbers x, y, z

Answer 1

(i)

To show: (x − y) + z = x − (y − z) for all numbers x, y, z.

We know that (x + y) + z = x + (y + z) for all numbers x, y, z. ….(i)

Now, (x − y) + z = (x + (−y)) + z

= x + ((−y) + z) (Using (i))

= x + (−y + z)

= x − (y − z)

Thus, (x − y) + z = x − (y − z) for all numbers x, y, z.

(ii)

To show: (x + y) − z = x + (y − z) for all numbers x, y, z.

We know that (x + y) + z = x + (y + z) for all numbers x, y, z. ….(i)

Now, (x + y) − z = (x + y) + (−z)

= x + (y + (−z)) (Using (i))

= x + (y − z)

Thus, (x + y) − z = x + (y − z) for all numbers x, y, z.