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-#+title:    Predicate Logic
-#+subtitle: (SE 212 Tutorial 5)
-#+author:   Amin Bandali
-#+email:    bandali@uwaterloo.ca
-#+date:     Wed Oct 9, 2019
-#+language: en
-#+options:  email:t num:t toc:nil \n:nil ::t |:t ^:t -:t f:t *:t <:t
-#+options:  tex:t  d:nil todo:t pri:nil tags:not-in-toc
-#+select_tags: export
-#+exclude_tags: noexport
-#+startup: beamer
-#+latex_class: beamer
-# #+latex_class_options: [bigger]
-#+latex_header: \setbeamercovered{transparent}
-#+latex: \setbeamertemplate{itemize items}[circle]
-#+beamer_color_theme: beaver
-
-* Today’s plan
-:PROPERTIES:
-:BEAMER_act: [<+->]
-:END:
-
-- do some semantics questions from homework 4
-- do some ND questions from homework 5
-
-* =h04q05=
-
-Provide a counterexample to show that the following argument is not
-valid and demonstrate that your answer is correct.
-
-#+begin_example
-forall y : M . exists x : N . p(g(x), y)
-|=
-exists z : M . p(z, z)
-#+end_example
-
-* =h04q05= \small{(cont’d)}
-
-#+begin_example
-Domain:
-    M = {m1, m2}
-    N = {n1, n2}
-
-Mapping:
-    Syntax  | Meaning
-    --------------------------
-      g(.)  |  G(n1) := m1
-            |  G(n2) := m2
-    --------------------------
-    p(., .) | P(m1, m1) := F
-            | P(m1, m2) := T
-            | P(m2, m1) := T
-            | P(m2, m2) := F
-#+end_example
-
-* =h04q05= \small{(cont’d)}
-
-#+begin_example
-Premise:
-      [forall y : M . exists x : N . p(g(x), y)]
-    = [exists x: N. p(g(x), ^m1)] AND
-      [exists x: N . p(g(x), ^m2)]
-    = (P(G(n1), m1) OR P(G(n2), m1)) AND
-      (P(G(n1), m2) OR P(G(n2), m2))
-    = (P(m1, m1) OR P(m2, m1)) AND
-      (P(m1, m2) OR P(m2, m2))
-    = (F OR T) AND (T OR F)
-    = T
-#+end_example
-
-* =h04q05= \small{(cont’d)}
-
-#+begin_example
-Conclusion:
-      [exists z: M . p(z, z)]
-    = P(m1, m1) OR P(m2, m2)
-    = F OR F
-    = F
-#+end_example
-
-* =h04q06=
-
-Express the following sentences in predicate logic. Use types in your
-formalization.  Is the set of formulas consistent?  Demonstrate that
-your answer is correct using the semantics of predicate logic.
-
-#+begin_example
-All programmer like some computers.
-Some programmers use MAC.
-Therefore, some people who like some computers use MAC.
-#+end_example
-
-* =h04q06= \small{(cont’d)}
-
-All programmer like some computers.\\
-Some programmers use MAC.\\
-Therefore, some people who like some computers use MAC.
-
-#+begin_example
-Formalization:
-    programmer(x) means x is a programmer
-    usesmac(x) means x uses MAC
-    likes(x, y) means x likes y
-
-forall x: Person . programmer(x) =>
-           exists y: Computer . likes(x, y),
-exists x: Person . programmer(x) & usesmac(x)
-|-
-exists x: Person .
-  (exists y: Computer . likes(x, y) & usesmac(x))
-#+end_example
-
-* =h04q06= \small{(cont’d)}
-
-These sentences are /consistent/.  Here is an interpretation in which
-all the formulas are T:
-
-#+begin_example
-Domain:
-    People = {John}
-    Computer = {MacPro}
-
-Mapping:
-    Syntax          | Meaning
-    -------------------------------------------
-    programmer(.)   | programmer(John) = T
-    likes(.,.)      | likes(John, MacPro) = T
-    usesmac(.)      | usesmac(John) = T
-#+end_example
-
-* =h04q06= \small{(cont’d)}
-
-#+begin_example
-formula 1:
-      [forall x: Person . programmer(x) =>
-       exists y: Computer . likes(x, y)]
-    = [programmer(^John) =>
-       exists y: Computer . likes(^John, y)]]
-    = programmer(John) IMP likes(John, MacPro)
-    = T IMP T
-    = T
-
-formula 2:
-      [exists x: Person . programmer(x) & usesmac(x)]
-    = programmer(John) AND usesmac(John)
-    = T AND T
-    = T
-#+end_example
-
-* =h04q06= \small{(cont’d)}
-
-#+begin_example
-formula 3:
-      [exists x: Person . (exists y: Computer .
-       likes(x, y) & usesmac(x))]
-    = [exists y: Computer .
-       likes(^John, y) & usesmac(^John)]
-    = likes(John, MacPro) AND usesmac(John)
-    = T AND T
-    = T
-#+end_example
-
-* =h05q01a=
-
-If the following arguments are valid, use natural deduction AND
-semantic tableaux to prove them; otherwise, provide a counterexample.
-
-#+begin_example
-forall x . s(x) | t(x),
-forall x . s(x) => t(x) & k(c, x),
-forall x . t(x) => m(x)
-|-
-m(c)
-where c is a constant
-#+end_example
-
-* =h05q01a= \small{(cont’d)}
-
-#+begin_example
-#check ND
-forall x . s(x) | t(x),
-forall x . s(x) => t(x) & k(c, x),
-forall x . t(x) => m(x)
-|-
-m(c)
-#+end_example
-
-* =h05q01a= \small{(cont’d)}
-
-#+begin_example
-1) forall x . s(x) | t(x) premise
-2) forall x . s(x) => t(x) & k(c, x) premise
-3) forall x . t(x) => m(x) premise
-4) s(c) | t(c) by forall_e on 1
-5) s(c) => t(c) & k(c, c) by forall_e on 2
-6) t(c) => m(c) by forall_e on 3
-7) case s(c) {
-    8) t(c) & k(c, c) by imp_e on 5, 7
-    9) t(c) by and_e on 8
-    10) m(c) by imp_e on 6, 9
-}
-11) case t(c) {
-    12) m(c) by imp_e on 6, 11
-}
-13) m(c) by cases on 4, 7-10, 11-12
-#+end_example
-
-* =h05q01b=
-
-Is this formula a tautology?
-
-#+begin_example
-|- (exists x . p(x)) => forall y . p(y)
-#+end_example
-
-* =h05q01b= \small{(cont’d)}
-
-No, this formula is not a tautology.  Interpretation:
-
-#+begin_example
-1) Domain = {a, b}
-
-2) Mapping:
-    Syntax  | Meaning
-    ----------------------
-    p(.)    | P(a) = T
-            | P(b) = F
-
-Conclusion:
-  [(exists x. p(x)) => forall y. p(y)]
-= (P(a) OR P(b)) IMP (P(a) AND P(b))
-= (T OR F) IMP (T AND F)
-= T IMP F
-= F
-#+end_example
-
-* =h05q01d=
-
-Is this argument valid?
-
-#+begin_example
-forall x . p(x) | q(x),
-forall x . !p(x)
-|-
-forall x . q(x)
-#+end_example
-
-* =h05q01d= \small{(cont’d)}
-
-#+begin_example
-#check ND
-
-forall x . p(x) | q(x), forall x . !p(x) |- forall x . q(x)
-
-1) forall x . p(x) | q(x) premise
-2) forall x . !p(x) premise
-3) for every xg {
-    4) p(xg) | q(xg) by forall_e on 1
-    5) case p(xg) {
-        6) !p(xg) by forall_e on 2
-        7) q(xg) by not_e on 5, 6
-    }
-    8) case q(xg) {}
-    9) q(xg) by cases on 4, 5-7, 8-8
-}
-10) forall x. q(x) by forall_i on 3-9
-#+end_example
-
-* Announcements
-
-- no tutorial next week (Oct 16) (reading week)
-- no tutorial the week after (Oct 23) (midterm marking)
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