2f:[] 2d:{"title":"Sets","cells":[{"id":0,"value":"# Sets","pinned":false,"hidden":false,"mode":"md"},{"id":26,"value":"md`# 1. What Is a Set? (Formal Definition)\n\nA **set** is a well-defined collection of distinct elements.\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":33,"value":"tex`A = \\{1,2,3\\}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":35,"value":"md`If an element x belongs to A, we write:`","pinned":false,"hidden":false,"mode":"ojs"},{"id":41,"value":"tex`x \\in A`","pinned":false,"hidden":false,"mode":"ojs"},{"id":43,"value":"md`If it does not:`","pinned":false,"hidden":false,"mode":"ojs"},{"id":45,"value":"tex`x \\notin A`","pinned":false,"hidden":false,"mode":"ojs"},{"id":47,"value":"md`# 2. Core Set Operations (Algebraic Notation)`","pinned":true,"hidden":false,"mode":"ojs"},{"id":49,"value":"tex`\nA = \\{1,2,3\\}\n`","pinned":true,"hidden":false,"mode":"ojs"},{"id":51,"value":"tex`\nB = \\{3,4,5\\}\n`\n","pinned":false,"hidden":false,"mode":"ojs"},{"id":53,"value":"md`## Union`\n","pinned":false,"hidden":false,"mode":"ojs"},{"id":55,"value":"tex`A \\cup B`\n","pinned":false,"hidden":false,"mode":"ojs"},{"id":57,"value":"tex`\nA \\cup B = \\{x \\mid x \\in A \\text{ OR } x \\in B\\}\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":59,"value":"tex`\nA \\cup B = \\{1,2,3,4,5\\}\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":69,"value":"md`## Intersection`","pinned":true,"hidden":false,"mode":"ojs"},{"id":71,"value":"tex`A \\cap B`","pinned":true,"hidden":false,"mode":"ojs"},{"id":73,"value":"tex`\nA \\cap B = \\{x \\mid x \\in A \\text{ AND } x \\in B\\}\n`","pinned":true,"hidden":false,"mode":"ojs"},{"id":75,"value":"tex`\nA \\cap B = \\{3\\}\n`","pinned":true,"hidden":false,"mode":"ojs"},{"id":77,"value":"md`## Difference`","pinned":true,"hidden":false,"mode":"ojs"},{"id":79,"value":"tex`A - B`","pinned":false,"hidden":false,"mode":"ojs"},{"id":81,"value":"tex`\nA - B = \\{x \\mid x \\in A \\text{ AND } x \\notin B\\}\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":83,"value":"tex`\nA - B = \\{1,2\\}\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":90,"value":"tex`A - B \\neq B - A`","pinned":false,"hidden":false,"mode":"ojs"},{"id":93,"value":"md`## Complement`\n","pinned":false,"hidden":false,"mode":"ojs"},{"id":95,"value":"tex`A^c = U - A`","pinned":false,"hidden":false,"mode":"ojs"},{"id":97,"value":"tex`\nA^c = \\{x \\in U \\mid x \\notin A\\}\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":101,"value":"md`## Symmetric Difference (Exclusive OR)`\n","pinned":false,"hidden":false,"mode":"ojs"},{"id":103,"value":"tex`A \\triangle B`\n","pinned":false,"hidden":false,"mode":"ojs"},{"id":105,"value":"tex`\nA \\triangle B = (A - B) \\cup (B - A)\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":107,"value":"tex`\nA \\triangle B = (A \\cup B) - (A \\cap B)\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":110,"value":"tex`\nA \\triangle B = \\{1,2,4,5\\}\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":118,"value":"md`# 3. Algebraic Laws of Sets`","pinned":false,"hidden":false,"mode":"ojs"},{"id":134,"value":"### Commutative Laws","pinned":false,"hidden":false,"mode":"md"},{"id":120,"value":"tex`A \\cup B = B \\cup A`","pinned":false,"hidden":false,"mode":"ojs"},{"id":128,"value":"tex`A \\cap B = B \\cap A`","pinned":false,"hidden":false,"mode":"ojs"},{"id":130,"value":"### Associative Laws","pinned":false,"hidden":false,"mode":"md"},{"id":132,"value":"tex`(A \\cup B) \\cup C = A \\cup (B \\cup C)`\n","pinned":false,"hidden":false,"mode":"ojs"},{"id":137,"value":"tex`(A \\cap B) \\cap C = A \\cap (B \\cap C)`","pinned":false,"hidden":false,"mode":"ojs"},{"id":141,"value":"### Distributive Laws","pinned":false,"hidden":false,"mode":"md"},{"id":138,"value":"tex`A \\cap (B \\cup C) = (A \\cap B) \\cup (A \\cap C)`","pinned":false,"hidden":false,"mode":"ojs"},{"id":143,"value":"tex`A \\cup (B \\cap C) = (A \\cup B) \\cap (A \\cup C)`","pinned":false,"hidden":false,"mode":"ojs"},{"id":145,"value":"tex`a(b+c) = ab + ac`","pinned":false,"hidden":false,"mode":"ojs"},{"id":149,"value":"### Identity Laws","pinned":false,"hidden":false,"mode":"md"},{"id":151,"value":"tex`A \\cup \\emptyset = A`","pinned":false,"hidden":false,"mode":"ojs"},{"id":152,"value":"tex`A \\cap U = A`\n","pinned":false,"hidden":false,"mode":"ojs"},{"id":156,"value":"tex`A \\cup U = U`","pinned":false,"hidden":false,"mode":"ojs"},{"id":161,"value":"tex`A \\cap \\emptyset = \\emptyset`\n","pinned":false,"hidden":false,"mode":"ojs"},{"id":163,"value":"### Idempotent Laws","pinned":false,"hidden":false,"mode":"md"},{"id":165,"value":"tex`A \\cup A = A`\n","pinned":false,"hidden":false,"mode":"ojs"},{"id":166,"value":"tex`(A \\cap B)^c = A^c \\cup B^c`\n","pinned":false,"hidden":false,"mode":"ojs"},{"id":207,"value":"### Complement Laws","pinned":false,"hidden":false,"mode":"md"},{"id":209,"value":"tex`A \\cup A^c = U`","pinned":false,"hidden":false,"mode":"ojs"},{"id":213,"value":"tex`A \\cap A^c = \\emptyset`","pinned":false,"hidden":false,"mode":"ojs"},{"id":215,"value":"tex`(A^c)^c = A`","pinned":false,"hidden":false,"mode":"ojs"},{"id":169,"value":"### De Morgan's Laws","pinned":false,"hidden":false,"mode":"md"},{"id":171,"value":"tex`(A \\cup B)^c = A^c \\cap B^c`\n","pinned":false,"hidden":false,"mode":"ojs"},{"id":172,"value":"tex`(A \\cap B)^c = A^c \\cup B^c`","pinned":false,"hidden":false,"mode":"ojs"},{"id":122,"value":"
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\n","pinned":false,"hidden":false,"mode":"html"},{"id":19,"value":"\n# 4. Translation to Python\n\n","pinned":false,"hidden":false,"mode":"md"},{"id":23,"value":"tex`\n\\begin{aligned}\nA \\cup B &\\;\\longrightarrow\\; \\texttt{A | B} \\\\[6pt]\nA \\cap B &\\;\\longrightarrow\\; \\texttt{A \\& B} \\\\[6pt]\nA - B &\\;\\longrightarrow\\; \\texttt{A - B} \\\\[6pt]\nA \\triangle B &\\;\\longrightarrow\\; \\texttt{A \\^ B} \\\\[6pt]\nA \\subseteq B &\\;\\longrightarrow\\; \\texttt{A.issubset(B)} \\\\[6pt]\nx \\in A &\\;\\longrightarrow\\; \\texttt{x in A}\n\\end{aligned}\n`","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W2f","stdlib":"1","titling":"auto"}