a+b=4a+4bc+d=4c+4dc+d+a+b=4a+4b+4c+4dc+d+a+b=4(c+d)+4(a+b)
Let
A=c+dB=a+b∴A+B=4A+4B
The above defines addition by a+b=4a+4b
Also,
Let s be a scalar
Therefore,
s(a+b)=4sa+4sb
The above defines scalar multiplication by a+b=4a+4b
a+b=4a+4bb+a=4b+4ab+a=a+b
Hence addition is commutative
a+b=4a+4b(a+b)+c=4(a+b)+4c=4a+4b+4ca+(b+c)=4a+4(b+c)=4a+4b+4c(a+b)+c=a+(b+c)
Hence addition is associative
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