# Algebra problem (from lang's basic mathematics)

• Oct 20th 2012, 04:57 AM
tvd89
Algebra problem (from lang's basic mathematics)
Hello everyone,

I've recently decided to try and brush up on my math by self-studying Lang's basic mathematics. There's an exercise there which reads: "Justify each step, using commutativity and associativity in proving the following identities". I've solved 8/10 problems but the last two I can't seem to figure out:

9. (X - Y) - (Z -W) = (X + W) - Y -Z

and:

10. (X - Y) - (Z -W) = (X - Z) + (W - Y)

Before the problems you're given the principle of commutativity and associativity, as well as the identity: "-(A + B) = - A - B", which I assume are all relevant in solving these problems.

If I replace all the variables with numbers I can see clearly that they are identical, for some reason however I cannot translate that into a series of abstract steps that proves that these are identical. Any help would be greatly appreciated!
• Oct 20th 2012, 05:56 AM
johnsomeone
Re: Algebra problem (from lang's basic mathematics)
#9 (X - Y) - (Z -W) = (X + W) - Y -Z

First: (X - Y) - (Z -W) = (X - Y) + ( -Z + W )
Uses: Associative & Distributive Laws and, repeatedly, that P + (-1)Q = P - Q (call that fact *).

(X - Y) - (Z -W)
= (X - Y) + (-1)(Z + (-1)W)............................(*) and (*)
= (X - Y) + ( (-1)(Z) + (-1)[(-1)(W)] ).............Distributive Law
= (X - Y) + ( -Z + [(-1)(-1)](W) )....................(*) and Associative Law (for Multiplication)
= (X - Y) + ( -Z + [1](W) ).............................Arithmetic: (-1)(-1) = 1
= (X - Y) + ( -Z + W )....................................1 is the unit for multiplcation

Second: (X - Y) + ( -Z + W ) = (X + W) - Y -Z
Uses: The above, the Laws, and the identity: -( Y + Z ) = -Y - Z

(X - Y) + ( -Z + W )
= (X - Y) + ( W - Z ).......................................Commutativ e Law (for addition)
= (X + (-1)Y) + ( W + (-1)Z )..........................(*) and (*)
= [ (X + (-1)Y) + (W) ] + (-1)Z........................Associative Law (for addition), treating (X + (-1)Y) as a single value.
= [ X + ( (-1)Y + W ) ] + (-1)Z........................Associative Law (for addition).
= [ X + ( W + (-1)Y ) ] + (-1)Z........................Commutative Law (for addition).
= [ ( X + W ) + ( (-1)Y ) ] + (-1)Z....................Associative Law (for addition).
= ( X + W ) + [ (-1)Y + (-1)Z ]........................Associative Law (for addition), treating (X+W) as a single value.
= ( X + W ) + (-1)[ Y + Z ].............................Distributive Law.
= ( X + W ) - ( Y + Z )....................................(*)
= ( X + W ) - Y - Z.........................................Given identity.
• Oct 20th 2012, 06:13 AM
bjhopper
Re: Algebra problem (from lang's basic mathematics)
Back in 1930 the rule was "When removing brackets preceded by a minus sign change the signsof the terms within the brackets" There is a 1 understood after the minus so -1 must be multiplied by each term
• Oct 20th 2012, 06:50 AM
tvd89
Re: Algebra problem (from lang's basic mathematics)
Thank you very much both. It wasn't explained that you could do that; once you point it out it seems obvious though. I'm glad it wasn't such a simple solution as I thought it would be, because I felt quite stupid to get stuck in the first chapter like that :)

Going to try to apply this to problem 10.
• Oct 20th 2012, 07:03 AM
tvd89
Re: Algebra problem (from lang's basic mathematics)
10. (X - Y) - (Z -W) = (X - Z) + (W - Y):

(X - Y) - (Z -W)
= (X - Y) + (-1)(Z + (-1)W)............................(*) and (*)
= (X - Y) + ( (-1)(Z) + (-1)[(-1)(W)] ).............Distributive Law
= (X - Y) + ( -Z + [(-1)(-1)](W) )....................(*) and Associative Law (for Multiplication)
= (X - Y) + ( -Z + [1](W) ).............................Arithmetic: (-1)(-1) = 1
= (X - Y) + ( -Z + W )....................................1 is the unit for multiplication
= (X - Y) + ( W - Z ).......................................Commutativ e Law (for addition)
...........................................
= (X - Z) + ( W - Y )....................................... Multiple instances of Commutative Law (for addition) to shuffle around the variables in the right order

I think this would be correct then.
• Oct 20th 2012, 08:17 AM
johnsomeone
Re: Algebra problem (from lang's basic mathematics)
Quote:

Originally Posted by tvd89
I'm glad it wasn't such a simple solution as I thought it would be, because I felt quite stupid to get stuck in the first chapter like that

The main idea is straight forward. A great deal of the work in that derivation was futzing with the minus signs. Had it been all plus, it would have be much shorter, and looked like this:

#9Alternate: (X + Y) + (Z + W) = (X + W) + ( Y + Z )

(X + Y) + (Z + W)
= (X + Y) + ( W + Z ).....................................Commutative Law (for addition)
= [ (X + Y) + W ] + Z.....................................Associative Law (for addition), treating (X + Y) as a single value.
= [ X + ( Y + W ) ] + Z...................................Associative Law (for addition).
= [ X + ( W + Y ) ] + Z...................................Commutative Law (for addition).
= [ ( X + W ) + Y ] + Z...................................Associative Law (for addition).
= ( X + W ) + ( Y + Z )...................................Associative Law (for addition), treating (X+W) as a single value.
• Oct 20th 2012, 08:26 AM
johnsomeone
Re: Algebra problem (from lang's basic mathematics)