Algebra 1 and pre-algebra worksheets are something that all would love to have been able to dominate in high school. Sometimes they do not dominate it there and that's when it could be a bit difficult.
More from Arthouse-Kuklite
More from Maths
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
You May Also Like
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