Chapter 1.4, Problem 23P

Use deductive reasoning to tell whether each statement is true or false. If it is false, give a counterexample.

For all real numbers p, q and $r,p-q-r=p-r-q$.

To determine

To find : Whether the statement is true or false.

Answer to Problem 23P

True

Explanation of Solution

Given statement: For all real numbers p,q and r, $p-q-r=p-r-q$

Justification:

Let p,q and r be any real numbers.

By commutative law of addition, we have

  $q+r=r+q$

Multiplying both sides of equation by -1, we get

  $-1\left(q+r\right)=-1\left(r+q\right)$

By distributive law of multiplication over addition, we get

  $-q-r=-r-q$

Adding p to both sides of equation, we get

  $p-q-r=p-r-q$

Therefore, given statement is true.