a = a
This can be proven mathematically. In fact, it's one of the main axioms of mathematics. The other key ones are associativity and commutativity.
a = a
(Subtract a from both sides).
a - a = a - a
Look at the right hand side. Let's evaluate that part, but keep the left hand side the same.
a - a = 0
In other words, if you ever take any value for a and subtract it from itself, the result is always 0. By consequence, we can 'reverse' this inequality to demonstrate that a = a.
We can extend our argument using "proof by contradiction".
"Suppose a = a is not a universally true statement."
That would mean that there exists a set of complex numbers (in the form of a + bi) that do not satisfy the property of a = a. (I'll use the complex set, C, as it just covers all the properties of numbers that we are interested in.)
If you can find one example for a that, when a - a is calculated, does not yield 0, that would prove that a does not equal a.
In the first section, however, we proved that there cannot be such an a.
Therefore, a = a.
I wanted to understand the difference between "selbstverständlich" and "offensichtlich," so I did a bit of research, which I will share in case it is helpful to others.
Both can mean "obvious/obviously". But "selbstverständlich" is more like "it can be taken for granted/it goes without saying/of course," where something can simply be assumed to be true. "Offensichtlich" means something more like "transparent/blatant/plainly", like you wouldn't necessarily have assumed it but it's clear as soon as you look at it. (Muttersprachler bitte beitragen, wenn ich Zwischentöne verpasst habe)
I guess one needs to be familiar with U.S. history to have seen or had an understanding of self-evident. "We hold these truths to be self-evident, that all men are created equal, and endowed by their Creator with certain unalienable rights..." It has the connotation of needing no explanation. It goes without saying. It is taken for granted. (by those who include themselves in the "we")
Ok, we all know selbstverständlich means self evident.. But no one says that. The true English translation is the word 'obvious', which is accepted by duolingo. So for 'es ist selbstverständlich' the true common English (American at least) is 'that's obvious' or just 'obviously'. What i want to know is, what is the real german transaltion? Is it selbstverständlich or is there another more common word or phrase? Dankeschön
I have seen the words in English and German to be somewhat similar, probably because they have common origins... But how do we deal with words like 'selbstverstandlich' which nowhere sound near to 'evident'. Please could someone brief me about, how do these words come into the language...
It means more like "it is self-evident"
In this word, it's pronounced like an "f".