The cancellation law of addition isn't necessary since we can do the following : By axiom 5 applied to 0 (that exists by axiom 4: 0+(-0)=0 but by addition commutativity, 0+(-0)=(-0)+0 but by axiom 4 (-0)+0=-0 therefore, we have by transitivity of "=" : 0=0+(-0)=(-0)+0=-0 and 0=-0.
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this real analysis series is so unique, is so simple but so amazing to see the entire real number sistem being constructed with just 10 axioms.
"how do we know 2 is not equal to 0" that's why i'm loving this series lol
Someone needs to make a video "ILIKEMATHPHYSICS out of context"
Brother 0 is the identity of addition. By definition 0+x=x for any x. In particular, 0+0=0, so 0 is its own inverse.
The cancellation law of addition isn't necessary since we can do the following :
By axiom 5 applied to 0 (that exists by axiom 4: 0+(-0)=0 but by addition commutativity, 0+(-0)=(-0)+0 but by axiom 4 (-0)+0=-0 therefore, we have by transitivity of "=" : 0=0+(-0)=(-0)+0=-0 and 0=-0.
Since you have already proven 0•x=0 and that (-1)•x=-x, we can immediately see
-0=(-1)•0=0
Great videos! Thanks for posting them! I really enjoy watching them
You could prove in a future video that Q is dense in R
Great job one more time 👏🏼👏🏼👏🏼
What is point of that?