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24-576: In 2011, some records were removed from the file. The data includes: In 2011, the following information was removed: The Death Master File is a subset of the Social Security Administration's Numident database file, computerized in 1961, which contains information about all Social Security numbers issued since 1936. The Death Master File is considered a public document under the Freedom of Information Act , and monthly and weekly updates of

48-416: 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 is a subset of B may also be expressed as B includes (or contains) A or A

72-911: 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

96-514: 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}

120-486: 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

144-461: 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 }

168-476: 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}

192-468: 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)}

216-803: 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

240-533: 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

264-498: 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 the statement A ⊆ B {\displaystyle A\subseteq B} by applying

SECTION 10

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288-726: 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

312-406: 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

336-405: 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

360-500: 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

384-542: The Death Master File is around 95 percent. Social Security Administration distributes the file via National Technical Information Service . In May 2013, the cost of a single download (with no weekly, monthly or quarterly annual subscription costs) was $ 1825. The Social Security Administration has estimated that about 16 million decedents were missing from the File, leading to government benefits being paid out improperly;

408-683: The file are sold by the National Technical Information Service of the U.S. Department of Commerce . Knowing that a patient died is important in many observational clinical studies and is important for medical research. It is also used by financial and credit firms and government agencies to match records and prevent identity fraud . The Death Master File, in its SSDI form, is also used extensively by genealogists . Lorretto Dennis Szucs and Sandra Hargraves Luebking report in The Source: A Guidebook of American Genealogy (1997) that

432-429: 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

456-515: The inability to buy or rent property, and mistaken accusations of identity theft . The Office of the Inspector General called the error rate "very low", but noted that "SSA’s erroneous death entries can lead to mistaken benefit terminations and cause severe financial hardship and distress to affected people. ... When errors like this occur, it can be a long and difficult process to resurrect your financial life." Subset In mathematics,

480-558: 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 )

504-995: 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}} )

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528-736: 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

552-415: The total amount of improper payments in 2014 was estimated at $ 124 billion. Conversely, the Social Security Administration estimates that roughly 12,000 living people are added to the File annually, potentially due to clerical error . Because the File is used widely for commercial purposes, an erroneous listing can lead to not only a cessation of government benefits, but also the freezing of bank accounts ,

576-542: The total number of deaths in the United States from 1962 to September 1991 is estimated at 58.2 million. Of that number, 42.5 million (73 percent) are found in the Death Master File. Other research published by the Social Security Administration in 2002 suggests that for most years since 1973, 93 percent to 96 percent of deaths of individuals aged 65 or older were included in the DMF. Today the number of deaths, at any age, reported to

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