‫ﺗﻤﺮﻳﻦ ﺳﺮﻱ ﺳﻮﻡ‬
‫ﻣﺪﺭﺱ‪ :‬ﺩﻛﺘﺮ ﺁﺑﺎﻡ‬
‫ﺳﺎﺧﺘﻤﺎﻥﻫﺎﻱ ﮔﺴﺴﺘﻪ‬
‫ﺩﺳﺘﻴﺎﺭ ﺁﻣﻮﺯﺷﻲ‪ :‬ﻫﺎﺩﻱ ﻳﺎﻣﻲ‬
‫‪ - 1‬ﺑﺮﺍﻱ ﻫﺮ ﻳﻚ ﺍﺯ ﮔﺰﺍﺭﻩﻫﺎﻱ ﺯﻳﺮ ﻳﻚ ﺟﺪﻭﻝ ﺍﺭﺯﺵ ﺑﺴﺎﺯﻳﺪ‪.‬‬
‫ﺍﻟﻒ( ‪[p ᴧ (p → q)] → q‬‬
‫ﺏ( )‪q ↔ (⌐p ᴠ ⌐q‬‬
‫ﺩﺭ ﺍﻳﻦ ﮔﺰﺍﺭﻩﻫﺎ ﺁﻳﺎ ﺭﺍﺳﺘﮕﻮﻫﺎﻳﻲ ﻭﺟﻮﺩ ﺩﺍﺭﻧﺪ؟‬
‫‪ - 2‬ﻓﺮﺽ ﻛﻨﻴﺪ ‪ s ،r ،q ،p‬ﻣﻌﺮﻑ ﮔﺰﺍﺭﻩﻫﺎﻱ ﺯﻳﺮ ﺑﺎﺷﻨﺪ‪:‬‬
‫‪ :p‬ﻣﻦ ﻧﻮﺷﺘﻦ ﺑﺮﻧﺎﻣﻪ ﻛﺎﻣﭙﻴﻮﺗﺮﻳﻢ ﺭﺍ ﻗﺒﻞ ﺍﺯ ﻧﺎﻫﺎﺭ ﺗﻤﺎﻡ ﻣﻲﻛﻨﻢ‪.‬‬
‫‪ :q‬ﻣﻦ ﺍﻣﺮﻭﺯ ﺑﻌﺪﺍﺯﻇﻬﺮ ﺗﻨﻴﺲ ﺑﺎﺯﻱ ﻣﻲﻛﻨﻢ‪.‬‬
‫‪ :r‬ﻫﻮﺍ ﺁﻓﺘﺎﺑﻲ ﺍﺳﺖ‪.‬‬
‫‪ :s‬ﺭﻃﻮﺑﺖ ﻛﻢ ﺍﺳﺖ‪.‬‬
‫ﮔﺰﺍﺭﻩﻫﺎﻱ ﺯﻳﺮ ﺭﺍ ﺑﻪ ﺻﻮﺭﺕ ﻧﻤﺎﺩﻱ ﺑﻨﻮﻳﺴﻴﺪ‪.‬‬
‫ﺍﻟﻒ( ﺍﮔﺮ ﻫﻮﺍ ﺁﻓﺘﺎﺑﻲ ﺑﺎﺷﺪ‪ ،‬ﻣﻦ ﺑﻌﺪﺍﺯﻇﻬﺮ ﺗﻨﻴﺲ ﺑﺎﺯﻱ ﺧﻮﺍﻫﻢ ﻛﺮﺩ‪.‬‬
‫ﺏ( ﺗﻤﺎﻡ ﻛﺮﺩﻥ ﻧﻮﺷﺘﻦ ﺑﺮﻧﺎﻣﻪﻱ ﻛﺎﻣﭙﻴﻮﺗﺮﻳﻢ ﻗﺒﻞ ﺍﺯ ﺑﻌﺪﺍﺯﻇﻬﺮ‪ ،‬ﺑﺮﺍﻱ ﺑﺎﺯﻱ ﺗﻨﻴﺲ ﺑﻌﺪﺍﺯﻇﻬﺮ ﺿﺮﻭﺭﻱ ﺍﺳﺖ‪.‬‬
‫ﺝ( ﻫﻮﺍ ﺁﻓﺘﺎﺑﻲ ﺍﺳﺖ ﻭ ﻣﻦ ﺑﻌﺪﺍﺯﻇﻬﺮ ﺗﻨﻴﺲ ﺑﺎﺯﻱ ﺧﻮﺍﻫﻢ ﻛﺮﺩ‪.‬‬
‫ﺩ( ﺭﻃﻮﺑﺖ ﻛﻢ ﻭ ﻫﻮﺍﻱ ﺁﻓﺘﺎﺑﻲ ﺑﺮﺍﻱ ﺍﻳﻨﻜﻪ ﺑﻌﺪﺍﺯﻇﻬﺮ ﺗﻨﻴﺲ ﺑﺎﺯﻱ ﻛﻨﻢ ﻛﺎﻓﻲ ﻫﺴﺘﻨﺪ‪.‬‬
‫‪ - 3‬ﺑﺎ ﻓﺮﺽ ﺍﻳﻨﻜﻪ ﻫﺮ ﻓﺮﺩ ﻳﺎ ﺭﺍﺳﺘﮕﻮ ﺍﺳﺖ ﻳﺎ ﺩﺭﻭﻏﮕﻮ‪ ،‬ﺭﺍﺳﺘﮕﻮ ﻳﺎ ﺩﺭﻭﻏﮕﻮ ﺑﻮﺩﻥ ﻫﺮ ﻓﺮﺩ ﺭﺍ ﺗﻌﻴﻴﻦ ﻛﻨﻴﺪ‪.‬‬
‫• ﻭﺿﻌﻴﺖ ﺍﻭﻝ‬
‫ﻣﻬﺮﺍﻥ ﻣﻲﮔﻮﻳﺪ‪» :‬ﻓﻘﻂ ﻣﻦ ﻭ ﺳﻌﻴﺪ ﺭﺍﺳﺖ ﻣﻲﮔﻮﻳﻴﻢ!«‬
‫ﻓﺮﻫﺎﺩ ﻣﻲﮔﻮﻳﺪ‪» :‬ﺳﻌﻴﺪ ﻳﻚ ﺩﺭﻭﻏﮕﻮﺳﺖ!«‬
‫ﺳﻌﻴﺪ ﻣﻲﮔﻮﻳﺪ‪» :‬ﻳﺎ ﻓﺮﻫﺎﺩ ﺭﺍﺳﺖ ﻣﻲﮔﻮﻳﺪ ﻭ ﻳﺎ ﻣﻬﺮﺍﻥ ﺩﺭﻭﻍ ﻣﻲﮔﻮﻳﺪ!«‬
‫• ﻭﺿﻌﻴﺖ ﺩﻭﻡ‬
‫ﻣﺮﻳﻢ ﻣﻲﮔﻮﻳﺪ‪» :‬ﻫﻴﭻ ﻭﻗﺖ ﺑﻪ ﻛﻴﻤﻴﺎ ﺍﻋﺘﻤﺎﺩ ﻧﻜﻨﻴﺪ!«‬
‫ﻛﻴﻤﻴﺎ ﻣﻲﮔﻮﻳﺪ‪» :‬ﻣﺮﻳﻢ ﻫﻤﻴﺸﻪ ﺩﺭﻭﻍ ﻣﻲﮔﻮﻳﺪ!«‬
‫ﻫﺪﻱ ﻣﻲﮔﻮﻳﺪ‪» :‬ﻣﺮﻳﻢ ﻭ ﻛﻴﻤﻴﺎ ﻫﺮ ﺩﻭ ﺩﺭﻭﻍﮔﻮ ﻫﺴﺘﻨﺪ!«‬
‫ﻧﺴﻴﻢ ﻣﻲﮔﻮﻳﺪ‪» :‬ﺑﻴﻦ ﻣﻦ ﻭ ﻣﺮﻳﻢ ﺣﺪﺍﻗﻞ ﻳﻜﻲ ﺍﺯ ﻣﺎ ﺩﺭﻭﻏﮕﻮﺳﺖ!‬
‫‪ - 4‬ﺩﺭ ﺍﻳﻦ ﺑﺮﻧﺎﻣﻪﻱ ‪ C++‬ﭼﻨﺪ ﺑﺎﺭ ﺗﺎﺑﻊ )‪ print(x‬ﺻﺪﺍ ﺯﺩﻩ ﻣﻲﺷﻮﺩ؟‬
‫‪ - 5‬ﺍﻟﻒ( ﺛﺎﺑﺖ ﻛﻨﻴﺪ ﻛﻪ‬
‫‪(⌐p ᴠ q) ᴧ (p ᴧ (p ᴧ q)) ↔ p ᴧ q‬‬
‫ﺏ( ﺩﻭﮔﺎﻥ ﻧﺘﻴﺠﻪﻱ ﻗﺴﻤﺖ )ﺍﻟﻒ( ﺭﺍ ﺑﻨﻮﻳﺴﻴﺪ‪.‬‬
‫‪ - 6‬ﺍﺳﺘﺪﻻﻝﻫﺎﻱ ﺩﺭﺳﺖ ﺭﺍ ﺛﺎﺑﺖ ﻛﻨﻴﺪ ﻭ ﺍﺳﺘﺪﻻﻝﻫﺎﻱ ﻏﻠﻂ ﺭﺍ ﺭﺩ ﻧﻤﺎﻳﻴﺪ‪.‬‬
‫ﺍﻟﻒ( ‪{(⌐p ᴠ q) → r, r → (s ᴠ r), (⌐s) ᴧ (⌐u), (⌐u) → (⌐r)} ˫ r‬‬
‫ﺏ( )‪{p ᴠ q, q ᴠ r, p → (r ᴧ s)} ˫ (⌐r) ᴠ (s‬‬
‫ﺝ(‬
‫‪{p → q, (⌐r) ᴠ s, p ᴠ r} ˫ (⌐q) → s‬‬
‫‪ - 7‬ﺍﺯ ﺍﻃﻼﻋﺎﺕ ﺭﻭﺑﻪﺭﻭ ﻛﺪﺍﻡ ﻳﻚ ﺍﺯ ﮔﺰﺍﺭﻩﻫﺎﻱ ﺯﻳﺮ ﻗﺎﺑﻞ ﺍﺳﺘﻨﺘﺎﺝ ﺍﺳﺖ؟ ﻣﺮﺍﺣﻞ ﺍﺳﺘﻨﺘﺎﺝ ﺧﻮﺩ ﺭﺍ ﻧﺸﺎﻥ ﺩﻫﻴﺪ‪.‬‬
‫)‪⌐K(X) ᴠ M(X‬‬
‫)‪⌐K(X) → ⌐Q(X‬‬
‫)‪L(X) ᴧ ⌐Q(X) → N(X‬‬
‫))‪⌐(N(X) ᴧ M(Y) ᴧ ⌐P(X, Y‬‬
‫)‪⌐Q(ali‬‬
‫)‪K(amir‬‬
‫)‪L(ali‬‬
‫)‪a : P(amir, ali‬‬
‫)‪b : Q(ali) ᴧ M(amir‬‬
‫)‪c : N(amir) ᴠ ⌐K(ali‬‬
‫)‪d: K(amir) → P(ali, amir‬‬
‫‪ - 8‬ﻓﺮﺽ ﻛﻨﻴﺪ ‪a ↑ b = ⌐a ᴧ b‬‬
‫ﺍﻟﻒ( ﮔﺰﺍﺭﻩﺍﻱ ﻣﻌﺎﺩﻝ ﺑﺎ ﮔﺰﺍﺭﻩ )‪ (⌐p → q) ᴠ (p ᴧ q ᴧ r‬ﺑﻨﻮﻳﺴﻴﺪ ﻛﻪ ﺩﺭ ﺁﻥ ﺗﻨﻬﺎ ﺍﺯ ﻋﻤﻠﮕﺮ ↑ ﺍﺳﺘﻔﺎﺩﻩ ﺷﺪﻩ ﺑﺎﺷﺪ‪.‬‬
‫)ﺑﺎﻟﻄﺒﻊ ﺍﺯ ﻧﻘﻴﺾ ﻫﻢ ﻧﻤﻲﺗﻮﺍﻧﻴﺪ ﺍﺳﺘﻔﺎﺩﻩ ﻛﻨﻴﺪ‪.‬‬
‫ﺏ( ﻓﺮﺽ ﻛﻨﻴﺪ ﮔﺰﺍﺭﻩ ‪ p‬ﻏﻠﻂ ﺑﺎﺷﺪ‪ .‬ﺁﻳﺎ ﻣﻲﺗﻮﺍﻥ ﺑﺎ ﺍﺳﺘﻔﺎﺩﻩ ﺍﺯ ﺗﻨﻬﺎ ﻛﺎﺭﺍﻛﺘﺮﻫﺎﻱ ﻣﺠﻤﻮﻋﻪ }↑ ‪ { ( , ) , p ,‬ﮔﺰﺍﺭﻩﺍﻱ‬
‫ﺳﺎﺧﺖ ﻛﻪ ﺩﺭﺳﺖ ﺑﺎﺷﺪ؟‬
‫‪ - 9‬ﺩﺭﺳﺘﻲ ﮔﺰﺍﺭﻩﻫﺎﻱ ﺯﻳﺮ ﺭﺍ ﺑﺮﺭﺳﻲ ﻛﻨﻴﺪ‪.‬‬
‫)‪• (a → c) ᴠ (b → c) → ((a ᴧ b) → c‬‬
‫))‪• ((a ᴧ b) → c) → ((a → c) ᴧ (b → c‬‬
‫)‪• (⌐p ↔ ⌐q) ↔ (p ↔ q‬‬
‫))‪• ((P ᴠ Q) ᴧ (⌐P ᴠ Q)) → ((P ᴧ ⌐Q) ᴠ (⌐P ᴧ Q‬‬
‫‪- 10‬ﮔﺰﺍﺭﻩ ﺯﻳﺮ ﺭﺍ ﺑﻪ ﺻﻮﺭﺕ ﻧﻤﺎﺩﻱ ﺑﻨﻮﻳﺴﻴﺪ‪ .‬ﺳﭙﺲ ﻳﺎ ﺑﺮﻫﺎﻧﻲ ﺑﺮﺍﻱ ﺍﺳﺘﺪﻻﻝ ﺑﻴﺎﻭﺭﻳﺪ ﻳﺎ ﻣﺜﺎﻟﻲ ﺑﺮﺍﻱ ﻧﻘﺾ!‬
‫»ﻣﻘﺎﻃﻌﻪ ﺍﻧﺠﺎﻡ ﺧﻮﺍﻫﺪ ﺷﺪ ﺍﮔﺮ ﻭ ﺗﻨﻬﺎ ﺍﮔﺮ ﭘﻨﺠﺮﻩﻫﺎﻱ ﺟﺪﻳﺪ ﺧﺎﻧﻪ ﺩﺭ ﻣﺎﻩ ژﻭﺋﻦ ﻧﺼﺐ ﺷﺪﻩ ﺑﺎﺷﺪ‪ .‬ﺍﮔﺮ ﭘﻨﺠﺮﻩﻫﺎﻱ ﺟﺪﻳﺪ ﺩﺭ‬
‫ﻣﺎﻩ ژﻭﺋﻦ ﻧﺼﺐ ﺷﻮﻧﺪ‪ ،‬ﺁﻧﮕﺎﻩ ‪ E‬ﻣﻲﺗﻮﺍﻧﺪ ﺩﺭ ﺍﻭﻝ ژﻭﺋﻴﻪ ﺑﻪ ﺧﺎﻧﻪ ﺟﺪﻳﺪﺵ ﻧﻘﻞ ﻣﻜﺎﻥ ﻛﻨﺪ‪ .‬ﺍﮔﺮ ‪ E‬ﻧﺘﻮﺍﻧﺪ ﺩﺭ ﺍﻭﻝ ژﻭﺋﻴﻪ ﻧﻘﻞ‬
‫ﻣﻜﺎﻥ ﻛﻨﺪ ﺁﻧﮕﺎﻩ ﺑﺎﻳﺪ ﺍﺟﺎﺭﻩﻱ ﻣﺎﻩ ژﻭﺋﻴﻪﻱ ﺁﭘﺎﺭﺗﻤﺎﻧﺶ ﺭﺍ ﺑﭙﺮﺩﺍﺯﺩ‪ .‬ﭘﻨﺠﺮﻩﻫﺎ ﺩﺭ ﻣﺎﻩ ژﻭﺋﻦ ﻧﺼﺐ ﺷﺪﻩﺍﻧﺪ ﻳﺎ ‪ E‬ﺑﺎﻳﺪ ﺍﺟﺎﺭﻩﻱ‬
‫ژﻭﺋﻴﻪﻱ ﺁﭘﺎﺭﺗﻤﺎﻧﺶ ﺭﺍ ﺑﭙﺮﺩﺍﺯﺩ‪ .‬ﺑﻨﺎﺑﺮﺍﻳﻦ‪ E ،‬ﻣﺠﺒﻮﺭ ﻧﺨﻮﺍﻫﺪ ﺑﻮﺩ ﻛﻪ ﺍﺟﺎﺭﻩﻱ ﻣﺎﻩ ژﻭﺋﻴﻪﻱ ﺁﭘﺎﺭﺗﻤﺎﻧﺶ ﺭﺍ ﺑﭙﺮﺩﺍﺯﺩ‪«.‬‬
‫‪- 11‬ﻧﻘﻴﺾ ﻋﺒﺎﺭﺕ ])‪ ∃y ∀x ∀z [P(x) ᴧ Q(x, y) → R(x, y, z)] ᴠ ∀y ∃x ∀z [Q(x, y) ᴠ S(x, y, z‬ﺭﺍ ﺑﻪ‬
‫ﺩﺳﺖ ﺁﻭﺭﻳﺪ‪ .‬ﺩﻗﺖ ﻛﻨﻴﺪ ﻛﻪ ﻋﻼﻣﺖ ﻧﻘﻴﺾ ﺑﺎﻳﺪ ﺗﻨﻬﺎ ﻗﺒﻞ ﺍﺯ ‪ predicate‬ﻫﺎ ﺑﻴﺎﻳﺪ‪.‬‬
‫‪- 12‬ﺍﻟﻒ( ﺁﻳﺎ ﺩﻭ ﮔﺰﺍﺭﻩ )‪ ∀y ∃x P(x, y‬ﻭ )‪ ∀x ∃y P(x, y‬ﺑﺎ ﻳﻜﺪﻳﮕﺮ ﻣﻌﺎﺩﻟﻨﺪ؟ ﭼﺮﺍ؟ ﺍﮔﺮ ﻧﻴﺴﺘﻨﺪ ﺁﻳﺎ ﻣﻲﺗﻮﺍﻥ ﻳﻜﻲ ﺭﺍ ﺍﺯ‬
‫ﺩﻳﮕﺮﻱ ﻧﺘﻴﺠﻪ ﮔﺮﻓﺖ؟‬
‫ﺏ( ﺁﻳﺎ ﺩﻭ ﮔﺰﺍﺭﻩ ))‪ ∀x (P(x) ᴠ Q(x‬ﻭ )‪ ∀x P(x) ᴠ ∀x Q(x‬ﻣﻌﺎﺩﻟﻨﺪ؟ ﭼﺮﺍ؟‬
‫ﺝ( ﺁﻳﺎ ﺩﻭ ﮔﺰﺍﺭﻩ ))‪ ∀x ∃y (P(x) ᴠ Q(y‬ﻭ )‪ ∀x P(x) ᴠ ∃x Q(x‬ﻣﻌﺎﺩﻟﻨﺪ؟ ﭼﺮﺍ؟‬
‫‪- 13‬ﺭﺍﺳﺘﻲ ﮔﺰﺍﺭﻩ ﺯﻳﺮ ﺭﺍ ﺍﺯ ﺭﺍﻩ ﺑﺮﻫﺎﻥ ﺧﻠﻒ ﺛﺎﺑﺖ ﻛﻨﻴﺪ‪ .‬ﺩﺭ ﺍﻳﻦ ﮔﺰﺍﺭﻩ‪ ،‬ﻋﺎﻟﻢ ﺳﺨﻦ ﺍﺯ ﺍﻋﺪﺍﺩ ﺻﺤﻴﺢ ﺗﺸﻜﻴﻞ ﺷﺪﻩ ﺍﺳﺖ‬
‫]])‪∀x ∀y [(3x + √2 y = 0) → [(x = 0) ᴧ (y = 0‬‬
‫‪- 14‬ﺍﻟﻒ( ﺩﻭﮔﺎﻥ ﮔﺰﺍﺭﻩﻱ ‪ (⌐p ᴧ ⌐q) ᴠ (T0 ᴧ p) ᴠ p‬ﺭﺍ ﺑﻴﺎﺑﻴﺪ‪.‬‬
‫ﺏ( ﺑﺎ ﺍﺳﺘﻔﺎﺩﻩ ﺍﺯ ﻗﻮﺍﻧﻴﻦ ﻣﻨﻄﻖ ﻧﺸﺎﻥ ﺩﻫﻴﺪ ﻛﻪ ﻧﺘﻴﺠﻪ ﻗﺴﻤﺖ )ﺍﻟﻒ( ﻣﻨﻄﻘﺎ ﻫﻢﺍﺭﺯ ﺑﺎ ‪ p ᴧ ⌐q‬ﺍﺳﺖ‪.‬‬
‫‪- 15‬ﺩﺭ ﺯﻧﺪﺍﻧﻲ ﮔﻴﺮ ﺍﻓﺘﺎﺩﻩﺍﻳﻢ ﻛﻪ ﺩﻭ ﺳﺮﺑﺎﺯ ﺍﺯ ﺩﻭ ﺩﺭ ﺁﻥ ﻣﺮﺍﻗﺒﺖ ﻣﻲﻛﻨﻨﺪ‪ ،‬ﻳﻚ ﺩﺭ ﺁﻥ ﺁﺯﺍﺩﻱ‪ ،‬ﻭ ﺩﺭ ﺩﻳﮕﺮ ﺍﻋﺪﺍﻡ ﺍﺳﺖ‪) .‬ﻫﺮ ﺳﺮﺑﺎﺯ‬
‫ﺩﻡ ﻳﻚ ﺩﺭ ﺍﻳﺴﺘﺎﺩﻩ ﺍﺳﺖ( ﻣﻲﺩﺍﻧﻴﻢ ﻛﻪ ﻳﻜﻲ ﺍﺯ ﺳﺮﺑﺎﺯﻫﺎ ﻫﻤﻴﺸﻪ ﺭﺍﺳﺖ ﻭ ﺩﻳﮕﺮﻱ ﻫﻤﻴﺸﻪ ﺩﺭﻭﻍ ﻣﻲﮔﻮﻳﺪ‪ ،‬ﺍﻣﺎ ﻧﻤﻲﺩﺍﻧﻴﻢ‬
‫ﻛﺪﺍﻣﻴﻚ ﺭﺍﺳﺘﮕﻮ ﻭ ﻛﺪﺍﻣﻴﻚ ﺩﺭﻭﻏﮕﻮ ﺍﺳﺖ‪ .‬ﺑﻪ ﻣﺎ ﺍﻳﻦ ﻓﺮﺻﺖ ﺩﺍﺩﻩ ﺷﺪﻩ ﺍﺳﺖ ﻛﻪ ﺗﻨﻬﺎ ﻳﻚ ﺳﻮﺍﻝ ﺍﺯ ﻳﻜﻲ ﺍﺯ ﺳﺮﺑﺎﺯﻫﺎ‬
‫ﺑﭙﺮﺳﻴﻢ ﻭ ﺳﭙﺲ ﻳﻜﻲ ﺍﺯ ﺩﺭﻫﺎ ﺭﺍ ﺍﻧﺘﺨﺎﺏ ﻛﻨﻴﻢ‪ .‬ﭼﻪ ﻛﺎﺭ ﻣﻲﺗﻮﺍﻥ ﻛﺮﺩ؟‬
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‫ﺍﮔﺮ ﻫﺮﮔﻮﻧﻪ ﺍﺑﻬﺎﻡ ﻳﺎ ﻣﺸﻜﻞ ﺩﺭ ﻣﻮﺭﺩ ﺗﻤﺮﻳﻦ ﺩﺍﺷﺘﻴﺪ ﺑﻪ ﺁﺩﺭﺱ ﺯﻳﺮ ﺑﻔﺮﺳﺘﻴﺪ‪.‬‬
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