??????3: Logic

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??????3Logic

• ??????? (Kuang-Chi Chen)
• chichen6_at_mail.tcu.edu.tw

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Binomial Theorem - review

• 3.2 ?????(The Binomial Theorem)
• ? x ? y ???,? n ?????,?
• (x y)n
• Note C(n, r) ????????(binomial coefficient).

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Application of Binomial

• ????????n gt 0
• a) C(n, 0) C(n, 1) C(n, 2) C(n, n)
• 2n,
• b) C(n, 0) C(n, 1) C(n, 2) C(n, 3)
  (-1)nC(n, n)
• 0?

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Multinomial Theorem

• 3.3 ?????(The Multinomial theorem)
• ????n?k,(x1 x2 x3 xk)n
  ????,x1n1x2n2x3n3xknk ????
• n!/(n1! n2! n3! nk!)
• ????? ni ???? ? 0 ni n, ?1 i
  k,n1 n2 n3 nk n?

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Proof of Multinomial Thm

• 3.3 ?????(The Multinomial Theorem)
• ????????,x1n1x2n2x3n3xknk ????? n ????? n1 ?
  x1 ??,???? ( n n1 ) ????? n2 ? x2 ??,,???
• ( n n1 n2 nk-1 ) nk ? xk ???
• n!/(n1! n2! n3! nk!) ? (?????)

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Logic??
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Logic

• 1. ??????????????,????????????????
• 1.1 ???? ? ????(The Basic Connectives and Truth
  Tables)
• ?? (statements) or ?? (propositions)??????,??????
  ??,??????????????? p, q, r, s????
• e.g., p?????????????? q2 8 10?

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Negation

• ??? or ???????,???????
• e.g., r????????! s ???,?????
• ??????(negation)p?q (q) ?
• e.g., p??????????????? q2 8 ? 10?

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Logical Connectives

• ?????????????(logical connectives)??????(compound
  statement)?
• a) Conjunction(??)pq,p ? q?
• b) Disjunction(??)p?q,p ? q?
• c) Implication(??)p?q,p ?? q?
• d) Bioconditional(???)p?q,p ???? q?

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Conjunction

• a) Conjunction(??)pq,p ? q?
• e.g., p??????????????
• q?????? 1/10 ?????
• pq ????????????? ? ?????? 1/10 ?????

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Disjunction

• b) Disjunction(??) p?q,p ? q?
• Inclusive(??)p q,p?q ????? ? ???????
• Exclusive(??)p q,p?q ????? ? ????????
• e.g., p?????????????? q?????? 1/10 ?????
• p?q????????????? ? ?????? 1/10 ?????

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Implication

• c) Implication(??) p?q,p ?? q?
• ? p ? q (if - then)?
• p ? q ???? or p ? q ??????
• q ? p ???? or q ? p ???????
• p ?? q ?
• e.g., p?????????????? q?????? 1/10 ?????
• p?q? ????????????? ? ?????? 1/10 ?????

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Bioconditional

• d) Bioconditional(???) p?q,p ???? q?( p iff q,p
  if and only if q )
• p ? q ??????? p ? q ???????? ?
• e.g., p?????????????? q?????? 1/10 ?????
• p?q????????????? ???? ?????? 1/10 ?????

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Truth Tables

• ? 0 ???,? 1 ???,?????
• Note p?q,?? p?q? ?? ?,???? ??????????? ????
  ?????? ????????

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More Truth Tables
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More Truth Tables
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Decision or Selection

• 1.2 ?-?-??(If-Then-Else)
• If p then q, else r? p ????? q,? p ????? r ?
• ?????,if-then-else ????????????,? Java ? C,p
  ??????????,? x gt 2???????????????????,? q?r ?????
  ??????

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Tautology Contradiction

• ??????????????????,??????(tautology,T0
  )??????????????????,??????(contradiction,F0 )?
• ?????,?? ?? ? ?? ???????? ?????(valid
  argument)?

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Logic Law????
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Logically Equivalent

• 2.1 Logically Equivalent
• ??????????????
• ????????(logically equivalent),?? s1? s2 ,?? ?
  ?? s1 ???,???? ?? s2 ??,? ?? s1 ???,???? ?? s2
  ???
• E.g., (p?q) ? (p?q)
• (p?q) ? (p?q) (q?p)
• (p?q) ? (p?q) (q?p)

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Example

• E.g., (p?q) ? (p?q)
• (p?q) ? (p?q) (q?p)
• (p?q) ? (p?q) (q?p)

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Distributive Law

• ?????,-(a b) (-a) (-b)
• ????? p?q?r,
• ???(Distributive Law)
• a) p(q?r) ? (pq)?(pr)
• b) p?(qr) ? (p?q)(p?r)?
• ??? a(b c) (ab) (ac),??????????(note?
  ?????????)

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DeMorgan Law

• DeMorgan Law
• 1) (p?q) ? (p)(q)
• 2) (pq) ? (p)?(q)
• ?? DeMorgan Law?

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Logic Law (part 1)
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Logic Law (part 2)
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Duality

• 2.2 ??(Duality)
• ??????? (1) ?????,?????
• ? s ????,? s ??????????,? s ???(dual)?? sd?
• E.g., s(p q)?(rT0),
• ? sd(p?q) (r?T0) ?
• Note? s ? sd ?,q ?????

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The Principle of Duality

• ????(Principle of Duality)
• ? s ? t ?????????,? s ? t,? sd ? td ?
• ???? (2)-(10) ,?????,??????????????? ???????
• More Details in Chapter 15

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Example 1

• E.g., pA ?????,
• q B ????????
• (p?q)
• ? A ?????,? B ????????
• (q?p)
• ? B????????,? A?????? ?

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Example 2

• E.g., pA ?????????????,
• q A ??????????
• (p?q)
• ? A ?????????????,? A ??????????
• (q?p)
• ? A ???????????,? A ?????????????? ?
• (q?p)

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