Sorry for being such a newbie on algebra.
From my understanding, algebra can be summarised as set with (closed) operation(s) occasionally endowed with the optional extra "relation". The operations are intuitionally generalised from real world beings (forgive my layman terms), instant examples include + and x, both represent abstractions of real world necessity.
When it comes to field of sets, however, it seems a bit redundant to have both disjunction and conjunction when one can be defined by the other with the help of complement.
I am by no means questioning the importance of the two operations, esp. in logic application, but it the co-existence of conjunction/disjunction definition just for convenience in expression?
Thanks in advance.
Yes, it's redundant, but it's highly convenient to have both, and another reason is that in our natural language we have both 'or' and 'and', so it really helps us to think about and make sense of what we're doing when doing boolean algebra.
In fact, we could just use the NAND (or just the NOR) ... but that would be really inconvenient, as we don't naturally think in terms of NAND, and as expressions would get long and complicated, let alone the manipulations thereof.
An analogy: I can say that the numeral $5$ is 'redundant', since I can express it using a strong of $1$'s, e.g. $5$ would be $11111$. But would you therefore favor using only the digit $1$ and represent all other numbers using this unary representation? Probably not, and for good reason!
Another analogy: I can program any computer with long strings of $0$'s and $1$'s. So, any computer language like C or Python is 'redundant'. So ... should we actually be programming in terms of writing down strings of $0$'s and $1$'s? Again, clearly not!