Symbols

์ง‘ํ•ฉ๋ก 

๋‹ค์Œ ๊ธฐํ˜ธ๋“ค์„ ์ด์šฉํ•˜์—ฌ ๋ช‡๊ฐ€์ง€์˜ ํŠน์ˆ˜ํ•œ ์ง‘ํ•ฉ์„ ๋‚˜ํƒ€๋‚ด๊ธฐ๋กœ ํ•œ๋‹ค.

  • $\emptyset$: ๊ณต์ง‘ํ•ฉemptyย set.
  • $\SetN$, $\SetZ_{\geq 0}$: ์ž์—ฐ์ˆ˜naturalย number ์ „์ฒด์˜ ์ง‘ํ•ฉ. ์ž์—ฐ์ˆ˜์˜ ์ตœ์†Œ ์›์†Œ๋Š” $0$์ธ ๊ฒƒ์œผ๋กœ ํ•œ๋‹ค. ์ฆ‰, $\SetN=\{0,1,2,3,\ldots\}$.
    • $1$ ์ด์ƒ์˜ ์ž์—ฐ์ˆ˜ ์ „์ฒด์˜ ์ง‘ํ•ฉ์„ ๋‚˜ํƒ€๋‚ด๋Š” ๊ฒฝ์šฐ์—๋Š” $\SetZ_{>0}$๊ณผ ๊ฐ™์ด ํ‘œ๊ธฐํ•˜๋Š” ๊ฒƒ์œผ๋กœ ํ•œ๋‹ค.
  • $\SetZ$: ์ •์ˆ˜integer ์ „์ฒด์˜ ์ง‘ํ•ฉ.
  • $\SetQ$: ์œ ๋ฆฌ์ˆ˜rationalย number ์ „์ฒด์˜ ์ง‘ํ•ฉ.
  • $\SetR$: ์‹ค์ˆ˜realย number ์ „์ฒด์˜ ์ง‘ํ•ฉ.
  • $\SetC$: ๋ณต์†Œ์ˆ˜complexย number ์ „์ฒด์˜ ์ง‘ํ•ฉ.

์ง‘ํ•ฉ์— ๊ด€ํ•ด์„œ๋Š” ๋‹ค์Œ๊ณผ ๊ฐ™์€ ๊ธฐํ˜ธ๋ฅผ ์‚ฌ์šฉํ•˜๊ธฐ๋กœ ํ•œ๋‹ค.

  • $x$๊ฐ€ ์ง‘ํ•ฉ $A$์˜ ์›์†Œelement์ผ ๋•Œ, $x\in A$์™€ ๊ฐ™์ด ๋‚˜ํƒ€๋‚ธ๋‹ค.
  • ์ง‘ํ•ฉ $A$๊ฐ€ ์ง‘ํ•ฉ $B$์˜ ๋ถ€๋ถ„์ง‘ํ•ฉsubset์ด๋ผ๋Š” ๊ฒƒ์„ $A\subset B$ ํ˜น์€ $B\supset A$์™€ ๊ฐ™์ด ๋‚˜ํƒ€๋‚ธ๋‹ค.
  • ์ง‘ํ•ฉ $A$๊ฐ€ ์ง‘ํ•ฉ $B$์˜ ์ง„๋ถ€๋ถ„์ง‘ํ•ฉ์ด๋ผ๋Š” ๊ฒƒ, ์ฆ‰, $A\subset B$์ธ ๋™์‹œ์— $A\neq B$์ธ ๊ฒƒ์€ $A\subsetneq B$, ํ˜น์€ $B\supsetneq A$์™€ ๊ฐ™์ด ๋‚˜ํƒ€๋‚ด๊ธฐ๋กœ ํ•œ๋‹ค.
  • ์ง‘ํ•ฉ $A$์˜ ๋†๋„cardinality๋ฅผ $|A|$์™€ ๊ฐ™์ด ๋‚˜ํƒ€๋‚ด๊ธฐ๋กœ ํ•œ๋‹ค. ๋‹ค๋งŒ, $A$๊ฐ€ ๋ฌดํ•œ์ง‘ํ•ฉ์ธ ๊ฒฝ์šฐ์—, $A$๊ฐ€ ๋ฌดํ•œ์ง‘ํ•ฉ์ด๋ผ๋Š” ๊ฒƒ์„ ๋‚˜ํƒ€๋‚ด๊ธฐ ์œ„ํ•˜์—ฌ, ๋†๋„์™€๋Š” ๊ด€๊ณ„์—†์ด $|A|=\infty$์™€ ๊ฐ™์€ ๊ธฐํ˜ธ๋ฅผ ์‚ฌ์šฉํ•˜๊ธฐ๋กœ ํ•œ๋‹ค.
  • ์ง‘ํ•ฉ $A$์˜ ๋ฉฑ์ง‘ํ•ฉpowerย set, ์ฆ‰, $A$์˜ ๋ถ€๋ถ„์ง‘ํ•ฉ ์ „์ฒด์˜ ์ง‘ํ•ฉ์„ $2^A$์™€ ๊ฐ™์ด ๋‚˜ํƒ€๋‚ธ๋‹ค.
  • ์ง‘ํ•ฉ $A, B$์˜ ๊ต์ง‘ํ•ฉintersection, ํ•ฉ์ง‘ํ•ฉunion์„ ๊ฐ๊ฐ $A\cap B$, $A\cup B$์™€ ๊ฐ™์ด ๋‚˜ํƒ€๋‚ธ๋‹ค. ์ง‘ํ•ฉ $I$๋ฅผ ์ธ๋ฑ์Šค๋กœ ํ•˜๋Š” ์ง‘ํ•ฉ์กฑ $(A_i)_{i\in I}$์˜ ๊ต์ง‘ํ•ฉ, ํ•ฉ์ง‘ํ•ฉ์€ ๊ฐ๊ฐ $\bigcap_{i\in I}A_i$, $\bigcup_{i\in I}A_i$์™€ ๊ฐ™์ด ๋‚˜ํƒ€๋‚ด๋Š” ๊ฒƒ์œผ๋กœ ํ•œ๋‹ค.
  • ์ง‘ํ•ฉ $A, B$์˜ ์ง๊ณฑdirectย product, ์งํ•ฉdirectย sum์„ ๊ฐ๊ฐ $A\times B$, $A\sqcup B$์™€ ๊ฐ™์ด ๋‚˜ํƒ€๋‚ธ๋‹ค. ์ง‘ํ•ฉ $I$๋ฅผ ์ธ๋ฑ์Šค๋กœ ํ•˜๋Š” ์ง‘ํ•ฉ์กฑ $(A_i)_{i\in I}$์˜ ์ง๊ณฑ, ์งํ•ฉ์€ ๊ฐ๊ฐ $\prod_{i\in I}A_i$, $\bigsqcup_{i\in I}A_i$์™€ ๊ฐ™์ด ๋‚˜ํƒ€๋‚ด๋Š” ๊ฒƒ์œผ๋กœ ํ•œ๋‹ค.