small brain teaser... enjoy

1-4 are false, 5 is true.

assuming they start as true, it's 1,2 and 4 false, 3 and 5 true.
assuming they start as false, it's the opposite

Assuming 1-5 are all 1t-5t, 1-4 will be false and 5 is true.

I think no solution
To put it this way, let's consider 1's statement: "2,3 are false"

Assume 1t, then 2f 3f.
Assume 1f, then 2t 3t.

As we can see, no matter 1's true/falseness, 1 and 3 will always be opposite. Either 1t 3f, or 1f 3t.

Now let's consider 5's statement: "1, 3 are true"

Assume 5t, then 1t 3t
Assume 5f, then 1f 3f

As we can see, no matter 5's true/falseness, 1 and 3 will always be the same, Either 1t 3t, or 1f 3f.

Thus we reach a contradiction >_< Thus no solution I guess.

if 1 is true, then 2 and 3 are false (as stated) => 1 is true and 3 is false (so far so good) and 4 is true, 5 is false (the reverse of 3) => 2 is true, 3 is false and 1 is false and 3 is true - contradiction with the initial presupposition

if 1 is false, then 2 and 3 are true => 1 is false, 3 is true and 4 is false, 5 is true => 2 is false, 3 is true (the reverse of 4) and 1 is true - contradicting the initial supposition

So 1 can't be true or false, therefore the exercise has no solution?

1: true
2: false
3: false
4: false
5: false

When you set, f.e., (1) to FALSE you get 3 optional REAL statusses:
-(2) FALSE, (3) TRUE
-(2) TRUE, (3) FALSE
-(2) TRUE, (3) TRUE

Maybe I'm doing it wrong, but it looks like the only one that never changes is statement 5 (by the time you read statement 3 it always says statement 5 is true, and it's the only statement that can change 5), but 5 can't be true because 1 says statement 2&3 are true/false (depending on whether it's been reversed) and statement 2 never agrees that both statement 1 and statement 3 are true.... just looking at it, I'd say that statements 2 or 3 can be true, but not both of them... and I'm not sure how to determine which if either of them are true. Statements 1, 4 and 5 look to be definitely false.

statement 1) 2 and 3 are false.

1=
2=f
3=f
4=
5=

statement 2) 1 is false, 3 is true.

1=f
2=f
3=ft
4=
5=

statement 3) 4 is false, 5 is true.

1=
2=f
3=f
4=ftf
5=t

statement 4) 2 is true, 3 is false.

1=
2=ft
3=ff
4=ftf
5=t

statement 5) 1 is true, 3 is true.

1=t
2=ft
3=fft
4=ftf
5=t

2 false cancel out, no matter how many true's, a leftover false will make the statment false.
end result:

1=t
2=f
3=t
4=t
5=t

statement 1) 2 and 3 are false.
statement 2) 1 is false, 3 is true.
statement 3) 4 is false, 5 is true.
statement 4) 2 is true, 3 is false.
statement 5) 1 is true, 3 is true.

1) says 2 and 3 are false.
2) can no longer say 1 is false and 3 is true. it now says 1 is true and 3 is false
3) can no longer say 4 is false and 5 is true. it now says 4 is true and 5 is false.
4) says 2 is true, and 3 is false.
5) can no longer say 1 is true, 3 is true. now says 1 is false, 3 is true.

remember: one statement saying another statement is true, therefore keeps the statement the same.

the curren't status of the statements are:

1=tf
2=ft
3=ffft
4=t
5=f

final answer:

1=f
2=f
3=f
4=t
5=f

statement 1) 2 and 3 are false.
statement 2) 1 is false, 3 is true.
statement 3) 4 is false, 5 is true.
statement 4) 2 is true, 3 is false.
statement 5) 1 is true, 3 is true.
Starting at 1 = T

T 1) 2 false, 3 false
F 2) 1 true, 3 false
F 3) 4 true, 5 false
T 4) 2 false, 3 true

go back to 3)

T 3) 4 false, 5 true
F 4) 2 true, 3 false

go back to 2)

T 2) 1 false, 3 true

go back to 1)

F 1) 2 true 3 true
T 2) 1 false, 3 true
T 3) 4 false, 5 true
F 4) 2 true, 3 false

Loop???

none are true:
I) If 1 is true:
1 -2 and 3 are false
2 -since 3 is false, 4 is true
3 -which means that 2 is true
4 -which contradicts line 1

II) If 2 is true:
1 -3 is true
2 -which means that 4 is false
3 -which means that 2 is false
4 -which contradicts the starting premise

III) If 3 is true:
1 -5 is true
2 -which means that 1 is true
3 -which has already been shown to generate a contradiction

IV) If 4 is true:
1 -2 is true
2 -which has already been shown to generate a contradiction

V) If 5 is true:
1 -1 is true
2 -which has already been shown to generate a contradiction

1-4 each say that various other statements are false. If any of 1-4 are false (i.e. its reverse is true), that will make a different statement true. We already showed that none of the statements can be true; therefore none of statements 1-4 can be false. If 5 is false, then 1 & 3 are false, but we just said that 1-4 cannot be false.

In short, all 5 statements are neither true nor false.

true
false
true
false
true

1:True
2:False
3:True
4:False
5:True

Consider S3:
1 ) If 3 is true, it says that 5 is true, which says that 1 is true which says that 3 is [i]false (contradiction in italics)

2 ) If [b]3 is false it means that (through full reversal) 4 is true which says that 2 is true which then says that [b]3 is true (contradiction in italics)

So you could never find a value for 3, so no solution



...right?

statement 1) 2 and 3 are false.
statement 2) 1 is false, 3 is true.
statement 3) 4 is false, 5 is true.
statement 4) 2 is true, 3 is false.
statement 5) 1 is true, 3 is true.

Assume 1 is true:
2 is false -> 1 is true, 3 is false
3 is false -> 4 is true, 5 is false
4 is true -> 2 is true
RAA: 1 is false
When 1 is false, 5 is false.
When 5 is false, 3 is false.
Therefore making 4 and 2 true.
However, 2 says that 3 is true

So there is no answer.
Unless you take the path that "not false" doesn't mean "true." Then you only look at the "true" values:

Assume 2 is true:
3 is true
5 is true
1 is true, which contradicts 2 being true

Assume 3 is true:
5 is true
1 is true
can work.

Assume 4 is true:
2 is true
3 is true
5 is true
1 is true, contradicting 2 being true

Assume 5 is true:
1 is true
3 is true
5 is true

Assume 1 is true:
nothing.

Therefore, 1, 3, 5.

Assume statement 1 is true, and the rest are false.
Statement 1 is valid because we're assuming both 2 and 3 are false.
Statement 2 is false because 1 is true, and says 2 is false
Statement 3 is false because 1 is true, and says 3 is false.
Statement 4 is false because it says 2 is true, but 2 is actually false.
Statement 5 is false because it says 3 is true, but 3 is actually false.


Yeah, it's all based on assumptions, but based on that assumption, everything checks out.

Proof that 5 is false by contradiction:
1) Assume 5 is true. This implies that 1 is true and 3 is true.
2) Since 1 is true, both 2 and 3 are false. But this is a contradiction; 5 said 3 is true. Thus, 5 must be false.

Now we know 5 is false. Let's find any statements that say 5 is true. Statement 3 says 5 is true. Thus, statement 3 must be false.

Now let's find statements which say 3 is true. Both 2 and 5 say 3 is true; thus, both 2 and 5 must be false (we already knew 5 was false).

Finally, we find any statements that say 2 is true. Statement 4 says 2 is true, thus 4 must be false.

So far, we have proved that 2,3,4, and 5 are all false. Let's look at statement 1. Statement 1 says both statements 2 and 3 are false. Hence, statement 1 must be true.

Hence, we have figured out the values of all five statements:

1 is true, and 2,3,4,5 are all false.

Order of Operation:

statement 1) 2 and 3 are false. 1T2F3F4T5T
statement 2) 1 is false, 3 is true. 1T2F3F4T5T (Works good so far)
statement 3) 4 is false, 5 is true. 1T2F3F4T5F (Good)
statement 4) 2 is true, 3 is false. 1T2T3F4T5F (Everythings Fine)
statement 5) 1 is true, 3 is true. 1F2T3F4T5F (A bit of switching)
1F2T3T4T5F (Starting from beginning again)
1F2T3T4T5F (After 2)
1F2T3T4F5T (After 3)


I quit, dinner time

statement 1) 2 and 3 are false.
statement 2) 1 is false, 3 is true.
statement 3) 4 is false, 5 is true.
statement 4) 2 is true, 3 is false.
statement 5) 1 is true, 3 is true.

statement 1) = OK
statement 2) = OK
statement 3) = OK
statement 4) = 3 reverses to <b>true</b>! 2 stays the same <b>false</b>
statement 5) = 3 stays the same and so does 1.

1 F
2 F
3 T
4 F
5 T

assume 5 is true. 5 says 1 is true and 3 are true. ok so look at 1. 1 says 3 is false. direct contradiction with 5 saying that 3 is true.
so 5 IS FALSE. and we can rule out anything else that says 5 is true.
therefore 3 IS FALSE because it claims so. and by chain combo logic, anything saying 3 is true is also false.
so 2 IS FALSE for claiming so. the only thing that can be true is either 1 or 4. and because 4 says 2 is true, which we just prove is totally bogus, 4 is also bogus by the bogus-transitive property.
1 IS TRUE.




Check: 1 says 2 and 3 are false.
2 claims than 1 is false and 3 is true. that is not consistnet with 1. checks out
3 claims that 4 is false and 5 is true. 5 is not true, so that is also not consistent with 1. checks out.

1 IS TRUE

FTFTF
run the statement from 1 to 5 once.

Givens:

1 ⇔ ¬2 ∧ ¬3
2 ⇔ ¬1 ∧ 3
3 ⇔ ¬4 ∧ 5
4 ⇔ 2 ∧ ¬3
5 ⇔ 1 ∧ 3

Let's test statement 1, by assuming 1 is true.

A. 1 ⇔ ¬2 ∧ ¬3; let's test the individual components ¬2 and ¬3

B. ¬2 ⇔ ¬(¬1 ∧ 3) ⇔ 1 ∨ ¬3

We are assuming 1 is true, so that holds. How about ¬3? (Note that we are still interested in ¬3 because of statement A, not statement B)

C. ¬3 ⇔¬(¬4 ∧ 5) ⇔ 4 ∨ ¬5

Let's test 4 and ¬5, then.

D. 4 ⇔ 2 ∧ ¬3

But we already have ¬2. So let's check the other term from C, ¬5

E. ¬5 ⇔ ¬(1 ∧ 3) ⇔ ¬1 ∨ ¬3

¬3 we already hold to be true, so E is satisfied, which satisfies C, and so on... so statement 1 is possibly TRUE.

So far, if we assume that 1 is true, we find a consistent result that:
1, ¬2, ¬3, ¬4, ¬5.

Bleh, too lazy to go through all the statements -_- someone else can give it a try

Hah, I guess I was lucky in choosing to start with 1.

Assume 5 is true.
3 and 1 are then true. But 1 says 3 is false.
So 5 must be false.
3 is then false.
so 2 is then false.
Then 4 is false.
1 is true.

TFFFF

1 true, everithing else is false, just need to find a contradiction, and with 5 is easy to find, but try and u will find it on 2,3 and 4 too

Only the first statement is true. Just start from Statement 5 and work your way up from that.

1, 2 and 5 are true
3 and 4 are false

Assume 1 is true..
2 is false
3 is false
4 is false
5 is false.

Assume 1 is false.
Assume 2 is true
3 is true
4 is false
5 is true

Assume 1 is false
Assume 2 is false
3 is then not true, that would make 2 true.
But 3 cannot be false, or 1 would be true.

So I think my first two satisfy.

1 and 5 true, the rest false

Assuming 1 = true, 2 and 3 contain elements of falseness.

4 options:

yes, yes
yes, no
no, yes,
no, no

if 2 is completely false, that means, 1 is true and 3 contains elements of falseness. ( which is what we want since it matches up with statement 1)

So far, we have established based on our 1st assumption,

1= completely true.
2= completely false.

If 3 contains elements of falseness, then there are only 3 possibilites. Since the given statement is
"statement 3) 4 is false, 5 is true." and that this is false,
Only remaining is:

yes, yes
yes, no
no, no

SKipping ahead for a second, we want to maintain the fact that

1= completely true.
2= completely false.

Thus, with statement 4,"statement 4) 2 is true, 3 is false." We want it to be false so that 2 will be false.

Since we have assumed 3 contains elements of falseness, the only remaining choice is (no, no) since (no, yes) is assumed false.

We now have:

1= completely true.
2= completely false.
3= Half True. (but still false)

Since statement 3 allows us to continue the assumption that statement 4 is false, that means we are able to conclude some information.

We know that the first part, 2=yes is false. What we don't know is whether the second part of the statement, 3 is false is true or not.
However, since we already established 3 = half true, we can say the 2nd part of statement 4 is true.

We now have:

1= completely true.
2= completely false.
3= Half True. (but still false)
4= Half True ( but still false)

Statement 5 says 1= true. We must assume this to be true since it is the basis for everything. However, based from statement 3, we had established statement 5 is false. That means 3=yes is incorrect. Meaning 3 is false. ( Which we have established )

Thus

1=true
2=false
3=false
4=false
5=false

Is what I think the answer is.

TFFFF