Batch 2 - Class 53 - Logic, Venn Diagrams

• (Waterloo - Problem 11) There is a secret hall at Hogwart’s with magical closets. When you open a closet door, you might find Harry Potter, or you might find Voldemort. Since Hogwarts is magic, it is possible that both doors contain Harry Potter, or both doors contain Voldemort. 
• (a) When you enter the secret hall, you encounter two closet doors. “One of the signs is true,” says Professor Snape,“and the other is false.”  

  i. First Door: Harry is behind this door. Voldemort is behind the other.ii. Second Door: Harry is behind one of these doors. Voldemort is behind the other.Who is behind each door? Answer: First door has Voldemart, and the second has Harry (Solve using truth tables)

• Who is behind each door? 

  Answer: Same - First door has Voldemart, and the second has Harry (Solve using truth tables)

• Expression Evaluation - Revision

  What rows correspond to not (A implies B)?Give a more complex formula such as evaluation of (A and not B or C or not (D and E and F)) for A, B, D, E true and C, F falseProve that (A implies B) is same as (not A or B)Prove that not (A or B) is same as not A and not B (De Morgan's Law)
• Knights and Knaves - Two people from the island, A and B

  A says "I am Knave and B is knight" - who is who (Draw truth table, with T and F signifying Knight and Knave)A says "Both of us are Knights" and B says "A is lying and 2+2=4" - who is who?A says "If B is a knight then I am a knave" - who is who?

• Items are members of sets. Example: Humans is a set of which you and me are members
• Overlapping sets imply that items inside that overlap are members of both
• Draw Venn diagrams for two sets A and B, if nothing else is known about them

  Establish correspondence to truth table of two statements, each being T or F
• Draw Venn diagram for "All humans are mortals" - correspondence with truth table
• If "Socrates is human", then what must it mean about his mortality?
• What is A or B, A and B, not A in Venn diagrams?
• What is A implies B in Venn diagrams? (Technically, it should be worded as "a is member of A implies, a is member of B")
• Draw possible Venn diagrams for " No A is B . Some C is A. " What can we say about C and B?

  Three cases - C and B can be disjoint, C can include all of B, or C can include part of B.
• Repeat the Venn diagram formulation of other examples above

• Let kids draw conclusions, and draw truth tables and venn diagrams for the following

  Some cats have no tails, All cats are mammals. What can we say about mammals having tails?Some mammals have no tailsAll horses have hooves; No humans have hooves. What can we say about humans being horses?No humans are horses
• Mistakes in deduction

  Some A are B; some B are C; what can we say about A being C?Some cats are black things; some black things are televisions. What can we say about cats being televisions?All A are B; All A are C; what can we say about relationship between B and CAll cats are feline; all cats are mammals. What can we say about mammals being feline?No A is B; No B is C; What can we say about relationship between A and C?No fish are dogs; No dogs can fly. What can we say about fish being able to fly?

• A real puzzle with truth tables - Bob, Ted,and Carol play a hat game. This particular game is played with 5 hats, three red, and two blue. Each person gets one hat put on his or her head. Bob can see Ted’s hat and Carol’s hat but can’t see his own. Ted can see Bob’s hat and Carol’s hat, but not his own. Carol cannot see any of the hats. The game is to see which player is the first to figure out what color hat he or she is wearing. Bob goes first. He says he can’t tell. Then Ted says he can’t tell. Carol now says she is wearing a red hat. How did she know?

  Draw the equivalent of truth table for this (instead of T and F, it will be color of hats - first draw all possibilities, then start to reduce possibilities)"Constraining the truth table" can provide answers