The problems of 3rd grade mathematics work sheets are experienced by many children and their parents are usually frustrated. There is no more, there is information about how to solve the problem of your Mathematics task for children ... My suggestions will help.
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OK, I may be guilty of a DoS attack attempt on mathematicians' brains here, so lest anyone waste too much precious brain time decoding this deliberately cryptic statement, let me do it for you. •1/15
First, as some asked, it is to be parenthesized as: “∀x.∀y.((∀z.((z∈x) ⇒ (((∀t.((t∈x) ⇒ ((t∈z) ⇒ (t∈y))))) ⇒ (z∈y)))) ⇒ (∀z.((z∈x) ⇒ (z∈y))))” (the convention is that ‘⇒’ is right-associative: “P⇒Q⇒R” means “P⇒(Q⇒R)”), but this doesn't clarify much. •2/15
Maybe we can make it a tad less abstruse by using guarded quantifiers (“∀u∈x.(…)” stands for “∀u.((u∈x)⇒(…))”): it is then “∀x.∀y.((∀z∈x.(((∀t∈x.((t∈z) ⇒ (t∈y)))) ⇒ (z∈y))) ⇒ (∀z∈x.(z∈y)))”. •3/15
Maybe a tad clearer again by writing “P(u)” for “u∈y” and leaving out the quantifier on y, viꝫ: “∀x.((∀z∈x.(((∀t∈x.((t∈z) ⇒ P(t)))) ⇒ P(z))) ⇒ (∀z∈x.P(z)))” [✯]. Now it appears as an induction principle: namely, … •4/15
… “in order to prove P(z) for all z∈x, we can assume, when proving P(z), that P(t) is already known for all t∈z∩x” (n.b.: “(∀z.(Q(z)⇒P(z)))⇒(∀z.P(z))” can be read “in order to prove P(z) for all z, we can assume Q(z) known when proving P(z)”). •5/15
\u2200x.\u2200y.((\u2200z.((z\u2208x) \u21d2 ((\u2200t.((t\u2208x) \u21d2 (t\u2208z) \u21d2 (t\u2208y)))) \u21d2 (z\u2208y))) \u21d2 (\u2200z.((z\u2208x) \u21d2 (z\u2208y))))
— Gro-Tsen (@gro_tsen) February 12, 2021
First, as some asked, it is to be parenthesized as: “∀x.∀y.((∀z.((z∈x) ⇒ (((∀t.((t∈x) ⇒ ((t∈z) ⇒ (t∈y))))) ⇒ (z∈y)))) ⇒ (∀z.((z∈x) ⇒ (z∈y))))” (the convention is that ‘⇒’ is right-associative: “P⇒Q⇒R” means “P⇒(Q⇒R)”), but this doesn't clarify much. •2/15
Maybe we can make it a tad less abstruse by using guarded quantifiers (“∀u∈x.(…)” stands for “∀u.((u∈x)⇒(…))”): it is then “∀x.∀y.((∀z∈x.(((∀t∈x.((t∈z) ⇒ (t∈y)))) ⇒ (z∈y))) ⇒ (∀z∈x.(z∈y)))”. •3/15
Maybe a tad clearer again by writing “P(u)” for “u∈y” and leaving out the quantifier on y, viꝫ: “∀x.((∀z∈x.(((∀t∈x.((t∈z) ⇒ P(t)))) ⇒ P(z))) ⇒ (∀z∈x.P(z)))” [✯]. Now it appears as an induction principle: namely, … •4/15
… “in order to prove P(z) for all z∈x, we can assume, when proving P(z), that P(t) is already known for all t∈z∩x” (n.b.: “(∀z.(Q(z)⇒P(z)))⇒(∀z.P(z))” can be read “in order to prove P(z) for all z, we can assume Q(z) known when proving P(z)”). •5/15
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A brief analysis and comparison of the CSS for Twitter's PWA vs Twitter's legacy desktop website. The difference is dramatic and I'll touch on some reasons why.
Legacy site *downloads* ~630 KB CSS per theme and writing direction.
6,769 rules
9,252 selectors
16.7k declarations
3,370 unique declarations
44 media queries
36 unique colors
50 unique background colors
46 unique font sizes
39 unique z-indices
https://t.co/qyl4Bt1i5x
PWA *incrementally generates* ~30 KB CSS that handles all themes and writing directions.
735 rules
740 selectors
757 declarations
730 unique declarations
0 media queries
11 unique colors
32 unique background colors
15 unique font sizes
7 unique z-indices
https://t.co/w7oNG5KUkJ
The legacy site's CSS is what happens when hundreds of people directly write CSS over many years. Specificity wars, redundancy, a house of cards that can't be fixed. The result is extremely inefficient and error-prone styling that punishes users and developers.
The PWA's CSS is generated on-demand by a JS framework that manages styles and outputs "atomic CSS". The framework can enforce strict constraints and perform optimisations, which is why the CSS is so much smaller and safer. Style conflicts and unbounded CSS growth are avoided.
Legacy site *downloads* ~630 KB CSS per theme and writing direction.
6,769 rules
9,252 selectors
16.7k declarations
3,370 unique declarations
44 media queries
36 unique colors
50 unique background colors
46 unique font sizes
39 unique z-indices
https://t.co/qyl4Bt1i5x
PWA *incrementally generates* ~30 KB CSS that handles all themes and writing directions.
735 rules
740 selectors
757 declarations
730 unique declarations
0 media queries
11 unique colors
32 unique background colors
15 unique font sizes
7 unique z-indices
https://t.co/w7oNG5KUkJ
The legacy site's CSS is what happens when hundreds of people directly write CSS over many years. Specificity wars, redundancy, a house of cards that can't be fixed. The result is extremely inefficient and error-prone styling that punishes users and developers.
The PWA's CSS is generated on-demand by a JS framework that manages styles and outputs "atomic CSS". The framework can enforce strict constraints and perform optimisations, which is why the CSS is so much smaller and safer. Style conflicts and unbounded CSS growth are avoided.
On the occasion of youtube 20k and Twitter 70k members
A small tribute/gift to members
Screeners
technical screeners - intraday and positional both
before proceeding - i have helped you , can i ask you so that it can help someone else too
thank you
positional one
run - find #stock - draw chart - find levels
1- Stocks closing daily 2% up from 5 days
https://t.co/gTZrYY3Nht
2- Weekly breakout
https://t.co/1f4ahEolYB
3- Breakouts in short term
https://t.co/BI4h0CdgO2
4- Bullish from last 5
intraday screeners
5- 15 minute Stock Breakouts
https://t.co/9eAo82iuNv
6- Intraday Buying seen in the past 15 minutes
https://t.co/XqAJKhLB5G
7- Stocks trading near day's high on 5 min chart with volume BO intraday
https://t.co/flHmm6QXmo
Thank you
A small tribute/gift to members
Screeners
technical screeners - intraday and positional both
before proceeding - i have helped you , can i ask you so that it can help someone else too
thank you
positional one
run - find #stock - draw chart - find levels
1- Stocks closing daily 2% up from 5 days
https://t.co/gTZrYY3Nht
2- Weekly breakout
https://t.co/1f4ahEolYB
3- Breakouts in short term
https://t.co/BI4h0CdgO2
4- Bullish from last 5
intraday screeners
5- 15 minute Stock Breakouts
https://t.co/9eAo82iuNv
6- Intraday Buying seen in the past 15 minutes
https://t.co/XqAJKhLB5G
7- Stocks trading near day's high on 5 min chart with volume BO intraday
https://t.co/flHmm6QXmo
Thank you