The ability to spot the distributive laws and the facility in using them are perhaps the points through which the fault lines run at the earliest stages between those of us who seem to be blessed with a head for algebraic manipulations and those who aren’t.
Briefly, the distributive law of multiplication over addition and multiplication over subtraction state:
a (b + c) = ab + ac
a (b – c) = ab – ac
The corresponding laws of reasoning are:
p or (q and r) is equivalent to p or q and p or r
p and (q or r) is equivalent to p and q or p and r
While factoring a polynomial of the kind:
(y^3) + 3(y^2) - 5y - 15 = (y^2)(y + 3) - 5(y + 3) = (y + 3)[(y^2) - 5]
we look for a common factor that can be removed or, if we can’t see one, we look for a pattern which might allow us to extract it. Likewise with statements of natural language. The statement:
Either I have the answer wrong or there is a printing error and the sum was never meant to work out.
is equivalent to:
I have the answer wrong or there is a printing error, and I have the answer wrong or the sum was never meant to work out.
Where the pattern is harder to spot, or the mind is more reluctant to accept it, is in a statement like:
If you fail an exam or you pay your own fees, then you know the price of education.
which translates as:
If you fail an exam then you know the price of education, and if you pay your own fees then you know the price of education.
This grouping or uncoupling as the case may be is a skill which must be picked up early and honed to perfection through constant practice. Memory, like with DeMorgan’s laws, is notoriously fickle, and just won’t hold the equivalences. Either you see them or you don’t, but I guarantee that the skill is not innate.
Briefly, the distributive law of multiplication over addition and multiplication over subtraction state:
a (b + c) = ab + ac
a (b – c) = ab – ac
The corresponding laws of reasoning are:
p or (q and r) is equivalent to p or q and p or r
p and (q or r) is equivalent to p and q or p and r
While factoring a polynomial of the kind:
(y^3) + 3(y^2) - 5y - 15 = (y^2)(y + 3) - 5(y + 3) = (y + 3)[(y^2) - 5]
we look for a common factor that can be removed or, if we can’t see one, we look for a pattern which might allow us to extract it. Likewise with statements of natural language. The statement:
Either I have the answer wrong or there is a printing error and the sum was never meant to work out.
is equivalent to:
I have the answer wrong or there is a printing error, and I have the answer wrong or the sum was never meant to work out.
Where the pattern is harder to spot, or the mind is more reluctant to accept it, is in a statement like:
If you fail an exam or you pay your own fees, then you know the price of education.
which translates as:
If you fail an exam then you know the price of education, and if you pay your own fees then you know the price of education.
This grouping or uncoupling as the case may be is a skill which must be picked up early and honed to perfection through constant practice. Memory, like with DeMorgan’s laws, is notoriously fickle, and just won’t hold the equivalences. Either you see them or you don’t, but I guarantee that the skill is not innate.
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