Thread: 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!

#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.

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

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.

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.

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.

Re: Your #10:
That looks good up to "1 is the unit for multiplication". Your next step, using the commutative law, isn't wrong, but also isn't the most direct approach to your ultimate goal.
Maybe try to work it out in detail from: (X - Y) + ( -Z + W ).
In other words, show:
(X - Y) + ( -Z + W ) = (X - Z) + (W - Y).