Math Is Fun Forum

ANon

Guest

proofs!

1)Let S be the collection of all statements of the form P.P.P.P.P.P.P where each '.' is either a ‘∨’ or a ‘∧’. (There are 6 squares, and 26 = 64 such statements in S.) For example, the
statement P ∨ P ∨ P ∧ P ∨ P ∧ P ∧ P s in S. (Note that ∧ is evaluated before ∨.) Show that all statements in S are logically equivalent.

2) Let T be the collection of all statements of the form P.P.P.P.P.P.P where each '.' is either a ‘∨’ or a ‘∧’, and there is EXACTLY ONE  that is a ‘⇒’. For example, the statement P ∨ P ∨ P ∧ P ∨ P ⇒ P ∧ P is in T. (Note that ∧ is evaluated before ∨, and ∨ is evaluated before ⇒.) Show that all statements in T are logically equivalent.

3) Create an example of a function f : R → R such that f(f(f(R))) = f(f(R)) does not equal f(R).

Bob

Administrator

Registered: 2010-06-20

Posts: 10,640

Re: proofs!

hi ANon

Welcome to the forum.

Assuming that all the Ps are the same then P^P = P and PVP = P.  It should be easy to prove that in Q1 all statements become P, and in Q2 they become P=>P

Q3.  If f is invertible then it won't work so we need a non-invertible function.  One way is to construct a function that leads to a constant after being applied twice.

eg:

f: If |x| ≥ 10        then f(x) = 10
   If 0 < |x| < 10  then f(x) 10.|x|
   If x = 0            then f(x) = 10

f(f(x))  = 10 for all x and f(f(f(x))) = 10 so these are equal.

But when x is between 0 and 10 , f(x) is not the same as f(f(x)).

Bob

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Children are not defined by school ...........The Fonz
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