- #1

happyprimate

- 7

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- Homework Statement
- Show that (a-b) + (c-d) = -(b+d) - (-a-c)

- Relevant Equations
- (a-b) + (c-d) = -(b+d) - (-a-c)

I would like to verify my solution for the following: (a-b) + (c-d) = -(b+d) - (-a-c)

Let p = (a-b) + (c-d). We need to show that p = -(b+d) - (-a-c)

(a-b) + (c-d) = (-b+a) + (c-d) By commutativity.

= (-b+a) + (-d+c) By commutativity.

= -b+[a+(-d+c)] By associativity.

= -b+[(a+(-d))+c] By associativity.

= -b+[(a-d)+c] By associativity.

= -b+[(-d+a)+c] By commutativity.

= -b+[-d+(a+c)] By associativity.

= (-b+(-d)) + (a+c) By associativity.

= (-b-d) + (a+c) By associativity.

= Factoring the negative, we get -(b+d) + (a+c).

a+c can be expressed as -(-a-c)

Therefore (a-b) + (c-d) = -(b+d) - (-a-c)

This is from Basic Mathematics by Serge Lang. Exercise 7 from chapter 1. Thank you.

Let p = (a-b) + (c-d). We need to show that p = -(b+d) - (-a-c)

(a-b) + (c-d) = (-b+a) + (c-d) By commutativity.

= (-b+a) + (-d+c) By commutativity.

= -b+[a+(-d+c)] By associativity.

= -b+[(a+(-d))+c] By associativity.

= -b+[(a-d)+c] By associativity.

= -b+[(-d+a)+c] By commutativity.

= -b+[-d+(a+c)] By associativity.

= (-b+(-d)) + (a+c) By associativity.

= (-b-d) + (a+c) By associativity.

= Factoring the negative, we get -(b+d) + (a+c).

a+c can be expressed as -(-a-c)

Therefore (a-b) + (c-d) = -(b+d) - (-a-c)

This is from Basic Mathematics by Serge Lang. Exercise 7 from chapter 1. Thank you.