Truthtellers and liars 2

You can try each statement and see if it works out. I think it's quicker to make a truthtable here.

Let A = Anne is a truthteller. Let B = Bob is a truthteller. Let C = Carl is a truthteller.

This means -A = Anna is a liar, and so on.

Now we can set up the truthtable.

Anne: "If Bob is a truthteller then Carl is a liar" is equivalent to A <=> (B => -C).

For something here to be "valid", it must have all the columns with statemsn to be "True". Now those are the only valid options, and we can see that there are two.

We can see that it is row 2 and row 5 where it is valid.

And now we just check to see what the values are for A,B,C for Row 2 and Row 5.

Row 2: Anne is a truthteller, Bob is a truthteller, Carl is a liar

Row 5: Anne is a liar, Bob is a truthteller, Carl is a truthteller.

We can see that Bob is always a truthteller. That Carl could be both be a liar or a truthteller. And that if Anna is a truthteller then Carl is a liar. And those are 3 correct statements.

Anne : If Bob is a truth teller, then Carl is a liar.

Conditional statements only have one way to be false, and that is when P's "Bob is a truth teller" is true and Q's "Carl is a liar" is false. So this means that Anne can only be a liar when the rest are truth tellers.

Bob : There's at least 1 truth teller in Anne or Carl.

From what we inferred from Anne's if-then statement, we get a result of either Anne be one of the truth tellers or all the others must be truth tellers themselves, so we know that there's at least a truth teller present between Anne and Carl, always and in anyways, so honest Bob is a confirmed truth teller.

Carl : There's at least 1 liar in Anne or Bob.

Since Bob is always a truth teller, Carl's statement depends solely on who Anne is. We found that Anne and Carl can only be opposite in nature, and this matches Anne's own statement earlier.

So, the only possibilities left are :
1) truth tellers Anne and Bob against a liar Carl, OR
2) truth tellers Carl and Bob against a lying Anne.

TRUE OR FALSE?

• Bob is always a truthteller ✓

• Anne is always a liar X

• Carl could both be a truthteller or a liar, we don't know ✓

• If Anne is a truthteller then Carl is a liar ✓

• If Bob is a truthteller, then Carl is a truthteller if and only if Anne is a truthteller X

• There are atleast 2 liars in the group X

Answer : 3 tick marks, so the number of true statements is 3.

Suppose Anne is a truthteller. Then Bob is a truthteller (he's telling the truth about Anne being a truthteller) and Carl is a liar (he's lying about one of the others being a liar). We note that in this case, Anne is telling the truth about Bob and Carl, so it's consistent.

Now suppose Anne is a liar. Then we can conclude Bob and Carl are truthtellers, as this is the only situation in which Anne's statement is false. It's easy to check that Bob's statement is true (Carl is a truthteller) and Carl's statement is true (Anne is a liar), so once again, we have a consistent set of statements.

We can conclude, as Anne is either a truthteller or a liar, that only these two situations are possible:

• Anne and Bob are truthtellers, Carl is a liar
• Anne is a liar, Bob and Carl are truthtellers.

It's then easy to see that

• Bob is always a truthteller
• Carl could both be a truthteller or a liar, we don't know
• If Anne is a truthteller then Carl is a liar

are the only $\boxed{3}$ of the given statements that are true.