Memoirs of an Endless Existence: the surprise that never was

Memoirs of an Endless Existence

"Memory's a wonderful thing if you don't have to deal with the past." - Julie Delpy in "Before Sunset (2004)"

28 February 2007

ANNOUNCEMENT: A teacher announces in the class that she will take a surprise quiz in the next week.

DEFINITIONS:
1) a: exactly one i.e. ((quiz on Xday)AND(quiz on Yday)) = FALSE [for all Xday and Yday]
2) surprise: it can never be determined with 100 % confidence on which day the quiz will be held i.e. (quiz on Xday) != TRUE [for any Xday]
3) in the next week: ((quiz on Monday)OR(quiz on Tuesday)OR(quiz on Wednesday)OR(quiz on Thursday)OR(quiz on Friday)) = TRUE

CLAIM 1: One of the student gets up and says that the surprise quiz cannot be on Friday, because

CLAIM 1.(i)
If ((quiz on Monday)OR(quiz on Tuesday)OR(quiz on Wednesday)OR(quiz on Thursday)) = TRUE
Then (quiz on Friday) = FALSE

PROOF 1.(i)

Case I: (quiz on Monday) = TRUE
==> ((quiz on Monday)AND(quiz on Friday)) = FALSE [exactly one quiz]
but since (quiz on Monday) = TRUE
hence, (quiz on Friday) = FALSE

Case II: (quiz on Tuesday) = TRUE
==> ((quiz on Tuesday)AND(quiz on Friday)) = FALSE [exactly one quiz]
but since (quiz on Tuesday) = TRUE
hence, (quiz on Friday) = FALSE

Case III: (quiz on Wednesday) = TRUE
==> ((quiz on Wednesday)AND(quiz on Friday)) = FALSE [exactly one quiz]
but since (quiz on Wednesday) = TRUE
hence, (quiz on Friday) = FALSE

Case IV: (quiz on Thursday) = TRUE
==> ((quiz on Thursday)AND(quiz on Friday)) = FALSE [exactly one quiz]
but since (quiz on Thursday) = TRUE
hence, (quiz on Friday) = FALSE

Case V: ((quiz on Monday = FALSE)AND(quiz on Tuesday = FALSE)AND(quiz on Wednesday = FALSE)AND(quiz on Thursday = FALSE)) = TRUE [for completeness of all the CASES]
Let's take complement of both the sides,
==> ((quiz on Monday)OR(quiz on Tuesday)OR(quiz on Wednesday)OR(quiz on Thursday)) = FALSE
But it contradicts with our assumption
Hence, Case I to Case IV are complete representation of the given assumption.
In Case I to Case IV, (quiz on Friday) = FALSE
Hence,
If ((quiz on Monday)OR(quiz on Tuesday)OR(quiz on Wednesday)OR(quiz on Thursday)) = TRUE
Then (quiz on Friday) = FALSE

CLAIM 1.(ii)
((quiz on Monday)OR(quiz on Tuesday)OR(quiz on Wednesday)OR(quiz on Thursday)) = TRUE always

PROOF 1.(ii)
Let's prove by contradiction.
Let's assume that,
((quiz on Monday)OR(quiz on Tuesday)OR(quiz on Wednesday)OR(quiz on Thursday)) = FALSE
But,
((quiz on Monday)OR(quiz on Tuesday)OR(quiz on Wednesday)OR(quiz on Thursday)OR(quiz on Friday)) = TRUE [as per the ANNOUNCEMENT]
Hence,
(quiz on Friday) = TRUE
But, this contradicts with our definition of "surprise"
Hence, this case is not possible

Combining both the claims, we get that (quiz on Friday) = FALSE always

Hence, we can update our definition of "in the next week"

DEFINITIONS(update):
3) in the next week: ((quiz on Monday)OR(quiz on Tuesday)OR(quiz on Wednesday)OR(quiz on Thursday)) = TRUE

CLAIM 2: Assuming CLAIM 1 to be true (and updated definitions), the quiz cannot be on Thursday

CLAIM 2.(i)
If ((quiz on Monday)OR(quiz on Tuesday)OR(quiz on Wednesday)) = TRUE
Then (quiz on Thursday) = FALSE

PROOF 2.(i)
[Similar to PROOF 1.(i)]

CLAIM 2.(ii)
((quiz on Monday)OR(quiz on Tuesday)OR(quiz on Wednesday)) = TRUE always

PROOF 2.(ii)
[Similar to PROOF 1.(ii)]

Combining both the claims, we get that (quiz on Thursday) = FALSE always

Continuing, we get
(quiz on Wednesday) = FALSE, and
(quiz on Tuesday) = FALSE

But, it was our initial assumption that
(quiz on Monday)OR(quiz on Tuesday)OR(quiz on Wednesday)OR(quiz on Thursday)OR(quiz on Friday) = TRUE

Hence,
(quiz on Monday) = TRUE
But, this contradicts with our definition of "surprise"
Hence, this case is not possible.

Hence, the ANNOUNCEMENT cannot be true.

Hence, no surprise quiz was held in the week :-)

Labels: comments, faith and temperance, intellectual, paradox, question, surprise quiz

posted by Rohit Agarwal

2 Comments:

At 09 December, 2007, SB said…

try reading
Hofstadter's Godel, Escher, Bach
Seems like yr kinda book.
At 09 December, 2007, SB said…

the teacher was told this.
she smiled and agreed.

on wednesday at 8 pm, she announced the test for wednesday.

the students were surprised!