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21-566: While technically they may be any subset of the much longer list of the Hundred Treasures , there is a combination that is most popular. Chinese Eight Treasures charms ( Traditional Chinese : 八寶錢; Simplified Chinese : 八宝钱; Pinyin : bā bǎo qián ), also known as the "Eight Precious Things charms" and the "Eight Auspicious Treasures charms", are coin amulets that depict the Eight Treasures. Those most commonly depicted on older charms are

42-514: Is less than y (an irreflexive relation ). Similarly, using the convention that ⊂ {\displaystyle \subset } is proper subset, if A ⊆ B , {\displaystyle A\subseteq B,} then A may or may not equal B , but if A ⊂ B , {\displaystyle A\subset B,} then A definitely does not equal B . Another example in an Euler diagram : The set of all subsets of S {\displaystyle S}

63-486: Is vacuously a subset of any set X . Some authors use the symbols ⊂ {\displaystyle \subset } and ⊃ {\displaystyle \supset } to indicate subset and superset respectively; that is, with the same meaning as and instead of the symbols ⊆ {\displaystyle \subseteq } and ⊇ . {\displaystyle \supseteq .} For example, for these authors, it

84-461: Is a valid inference rule . It states that if ⊢ P ( x ) {\displaystyle \vdash \!P(x)} has been derived, then ⊢ ∀ x P ( x ) {\displaystyle \vdash \!\forall x\,P(x)} can be derived. The full generalization rule allows for hypotheses to the left of the turnstile , but with restrictions. Assume Γ {\displaystyle \Gamma }

105-476: Is a set of formulas, φ {\displaystyle \varphi } a formula, and Γ ⊢ φ ( y ) {\displaystyle \Gamma \vdash \varphi (y)} has been derived. The generalization rule states that Γ ⊢ ∀ x φ ( x ) {\displaystyle \Gamma \vdash \forall x\,\varphi (x)} can be derived if y {\displaystyle y}

126-584: Is a subset of B may also be expressed as B includes (or contains) A or A is included (or contained) in B . A k -subset is a subset with k elements. When quantified, A ⊆ B {\displaystyle A\subseteq B} is represented as ∀ x ( x ∈ A ⇒ x ∈ B ) . {\displaystyle \forall x\left(x\in A\Rightarrow x\in B\right).} One can prove

147-468: Is an unsound deduction. Note that Γ ⊢ ∀ y φ ( y ) {\displaystyle \Gamma \vdash \forall y\,\varphi (y)} is permissible if y {\displaystyle y} is not mentioned in Γ {\displaystyle \Gamma } (the second restriction need not apply, as the semantic structure of φ ( y ) {\displaystyle \varphi (y)}

168-803: Is called its power set , and is denoted by P ( S ) {\displaystyle {\mathcal {P}}(S)} . The inclusion relation ⊆ {\displaystyle \subseteq } is a partial order on the set P ( S ) {\displaystyle {\mathcal {P}}(S)} defined by A ≤ B ⟺ A ⊆ B {\displaystyle A\leq B\iff A\subseteq B} . We may also partially order P ( S ) {\displaystyle {\mathcal {P}}(S)} by reverse set inclusion by defining A ≤ B  if and only if  B ⊆ A . {\displaystyle A\leq B{\text{ if and only if }}B\subseteq A.} For

189-533: Is equivalent to A ⊆ B , {\displaystyle A\subseteq B,} as stated above. If A and B are sets and every element of A is also an element of B , then: If A is a subset of B , but A is not equal to B (i.e. there exists at least one element of B which is not an element of A ), then: The empty set , written { } {\displaystyle \{\}} or ∅ , {\displaystyle \varnothing ,} has no elements, and therefore

210-726: Is not being changed by the substitution of any variables). Prove: ∀ x ( P ( x ) → Q ( x ) ) → ( ∀ x P ( x ) → ∀ x Q ( x ) ) {\displaystyle \forall x\,(P(x)\rightarrow Q(x))\rightarrow (\forall x\,P(x)\rightarrow \forall x\,Q(x))} is derivable from ∀ x ( P ( x ) → Q ( x ) ) {\displaystyle \forall x\,(P(x)\rightarrow Q(x))} and ∀ x P ( x ) {\displaystyle \forall x\,P(x)} . Proof: In this proof, universal generalization

231-406: Is not mentioned in Γ {\displaystyle \Gamma } and x {\displaystyle x} does not occur in φ {\displaystyle \varphi } . These restrictions are necessary for soundness. Without the first restriction, one could conclude ∀ x P ( x ) {\displaystyle \forall xP(x)} from

SECTION 10

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252-405: Is true of every set A that A ⊂ A . {\displaystyle A\subset A.} (a reflexive relation ). Other authors prefer to use the symbols ⊂ {\displaystyle \subset } and ⊃ {\displaystyle \supset } to indicate proper (also called strict) subset and proper superset respectively; that is, with

273-576: The China Time-honored Brand list by the Chinese Ministry of Commerce in 2011. Subset In mathematics, a set A is a subset of a set B if all elements of A are also elements of B ; B is then a superset of A . It is possible for A and B to be equal; if they are unequal, then A is a proper subset of B . The relationship of one set being a subset of another is called inclusion (or sometimes containment ). A

294-548: The Museum of Ethnography, Sweden has the inscription Chángmìng fùguì jīnyù mǎntáng (長命富貴金玉滿堂) which could be translated as "longevity, wealth and honour - may gold and jade fill your house (halls)". In Zhangzhou , Fujian , China there is a company named Babao seal paste which is named after the Eight Treasures. Babao seal paste was added to the National Intangible Cultural Heritage List in 2008. and

315-481: The ceremonial ruyi (sceptre), coral , lozenge , rhinoceros horns , sycees , stone chimes, and flaming pearl. Eight Treasures charms can alternatively display the eight precious organs of the Buddha's body, the eight auspicious signs, various emblems of the eight Immortals from Taoism , or eight normal Chinese character. They often have thematic inscriptions. For example a Chinese eight treasures charm on display at

336-500: The k -tuple from { 0 , 1 } k , {\displaystyle \{0,1\}^{k},} of which the i th coordinate is 1 if and only if s i {\displaystyle s_{i}} is a member of T . The set of all k {\displaystyle k} -subsets of A {\displaystyle A} is denoted by ( A k ) {\displaystyle {\tbinom {A}{k}}} , in analogue with

357-429: The hypothesis P ( y ) {\displaystyle P(y)} . Without the second restriction, one could make the following deduction: This purports to show that ∃ z ∃ w ( z ≠ w ) ⊢ ∀ x ( x ≠ x ) , {\displaystyle \exists z\,\exists w\,(z\not =w)\vdash \forall x\,(x\not =x),} which

378-558: The notation for binomial coefficients , which count the number of k {\displaystyle k} -subsets of an n {\displaystyle n} -element set. In set theory , the notation [ A ] k {\displaystyle [A]^{k}} is also common, especially when k {\displaystyle k} is a transfinite cardinal number . Universal generalization In predicate logic , generalization (also universal generalization , universal introduction , GEN , UG )

399-995: The power set P ⁡ ( S ) {\displaystyle \operatorname {\mathcal {P}} (S)} of a set S , the inclusion partial order is—up to an order isomorphism —the Cartesian product of k = | S | {\displaystyle k=|S|} (the cardinality of S ) copies of the partial order on { 0 , 1 } {\displaystyle \{0,1\}} for which 0 < 1. {\displaystyle 0<1.} This can be illustrated by enumerating S = { s 1 , s 2 , … , s k } , {\displaystyle S=\left\{s_{1},s_{2},\ldots ,s_{k}\right\},} , and associating with each subset T ⊆ S {\displaystyle T\subseteq S} (i.e., each element of 2 S {\displaystyle 2^{S}} )

420-736: The same meaning as and instead of the symbols ⊊ {\displaystyle \subsetneq } and ⊋ . {\displaystyle \supsetneq .} This usage makes ⊆ {\displaystyle \subseteq } and ⊂ {\displaystyle \subset } analogous to the inequality symbols ≤ {\displaystyle \leq } and < . {\displaystyle <.} For example, if x ≤ y , {\displaystyle x\leq y,} then x may or may not equal y , but if x < y , {\displaystyle x<y,} then x definitely does not equal y , and

441-907: The statement A ⊆ B {\displaystyle A\subseteq B} by applying a proof technique known as the element argument : Let sets A and B be given. To prove that A ⊆ B , {\displaystyle A\subseteq B,} The validity of this technique can be seen as a consequence of universal generalization : the technique shows ( c ∈ A ) ⇒ ( c ∈ B ) {\displaystyle (c\in A)\Rightarrow (c\in B)} for an arbitrarily chosen element c . Universal generalisation then implies ∀ x ( x ∈ A ⇒ x ∈ B ) , {\displaystyle \forall x\left(x\in A\Rightarrow x\in B\right),} which

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