Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °

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SteveB
2013-05-21 21:59:13

I agree with Bob that the implies interpretation is the most sensible.
If we are going to think of H(x) as meaning
"a person x put through a function that outputs true if x is happy and false if x is not happy"
then we need the logical connective of implies which has a boolean item on both sides.

Suppose H(x) is true if x is a member of the set H of all happy people and false otherwise.
Now let T(x) be true similarly if x is a member of set T of all theatre goers (and false otherwise).

Now let H(x) => T(x) using the interpretation using Bob's method.
Compare with "H is a subset of T".

Let us make up a set of four people: {person1, person2, person3, person4}
Let H = {person1, person2}
Let T = {person1, person2, person3}

H is a subset of T. If you are in H you must be in T.
If you are in T then you are not always in H.

Does this mean that in terms of the way I have defined them "H is a subset of T" means the same as "H(x) implies T(x)"?
As far as I know it does. However for domain reasons, if we are dealing with logic, the implies definition is essential.
You could redefine the whole exercise using set operators rather that boolean operators using Union, Intersection
and Complement from the universal set of all people.
These could replace OR, AND and NOT. Subset would replace implies... (you get the general idea).

{7/3}
2013-05-21 21:16:12

I believe these are the answer:
a2  b5  c1  d6  e8  f4  g7  h3.

bob bundy
2013-05-21 20:21:33

hi mrpace and Steve

I think the U rotated symbol is probably a

That is \supset

Q8 then reads " for all x, being in T implies being in H and in Q"

There is certainly a sentence version of that.

An 'and' can be made using \wedge

You can use + for 'or' (although I don't like it as it looks like 'and' to me)

And finally, for not you can use \tilde{}

If you put something inside the curly brackets the squiggle is put over the something

eg

Hope that helps,

Bob

SteveB
2013-05-21 19:46:20

#### mrpace wrote:

h)there is at least one happy theatre goer who is quiet.
3) E›x((Hx ^ Tx) ^ Qx)

With the statement (h) this to my mind reads "There exists at least one person x who is (happy and a theatre goer and is quiet)".

The statement (3) seems to be the compact form of this.

I will try to help with a bit of LaTeX which can be done using math tags in square brackets, and using things like
\exists
\forall
I am not sure how to do a proper "and" symbol in LaTeX. The same applies to "or" and "not". I have used text for these.

#### Code:

$\exists x \text{ such that } H(x) \text{ and } T(x) \text { and } Q(x)$

When you have used the ~ symbol does this mean "not" or "a negation of" ?
Try comparing (2) with (a) for example.

U rotated clockwise half a turn

I wonder whether this means "is a subset of" or "is a proper subset of". (a "proper subset" of means "smaller subset" of)
Example: Let a set be {1,2,3} a subset would also include the set itself. A proper subset would have to be smaller.
{1,2} is a proper susbset of {1,2,3} (and also a subset).
{1,2,3} is a subset of {1,2,3} (but is not a proper subset).
Try comparing (7) and (g).
Are we saying "not(the set of happy people is a proper subset of the set of theatre goers)" ?

bob bundy
2013-05-21 17:02:13

hi mrpace

Sorry but your symbolism isn't making much sense to me.  What are the x symbols ?

Suggest you have a look first at

http://en.wikipedia.org/wiki/List_of_logic_symbols

and then 'translate your questions into Latex:

http://www.artofproblemsolving.com/Wiki … eX:Symbols

There are some other sites showing what code to use too (google Latex symbols)

Then I'll try to help with your problems.

Bob

{7/3}
2013-05-21 14:26:07

I'm not very good at english,do you want to match the two kinds of statements?

mrpace
2013-05-21 09:30:03

given that 'H' means "is happy", Q means "is quiet" and T means "is a theatre-goer", match the lists when the universe of discourse is people.

a)someone is not happy
b)someone happy is not quiet
c)not everyone is a theatre goer
d)no theatre goer is quiet
e)every theatre goer is both happy and quiet
f)if no theatre goer is happy, then some non theatre goers are quiet
g)not everyone who is happy is a theatre goer
h)there is at least one happy theatre goer who is quiet.

Note: let V=for all, ^=and, E=there exists, U=a U rotated clockwise half a turn (not even sure what it means.

1) ~VxTx
2) E›x~Hx
3) E›x((Hx ^ Tx) ^ Qx)
4)Vx(Tx U ~Hx) U E›x(~Tx ^ Qx)
5)›Ex(Hx ^ ~Qx)
6)~E›x(Tx ^ Qx)
7)~Vx(Hx U Tx)
8)Vx(Tx U (Hx ^ Qx))

yea good luck, anything is appreciated.