Is there any example of Two logically equivalent sentences that together are an inconsistent set?

Question

In my textbook it say this is true. But I do not see how. How can something be inconsient if they both have the same truth value.

Answer 1

Inconsistent simply means if you assume both are true, you can get a contradiction.

If each are a contradiction to begin with, then they are equivalent but inconsistent.

Being inconsistent together doesn't mean they have to be each be consistent by themselves.

Answer 2

Saying that a set of statements is inconsistent means that there is no way they can all be true simultaneously.

Saying that two statements are logically equivalent means that they invariably have the same truth value as each other.

So, suppose you have two statements, each of which is always false (these are called contradictions). For example, if you are talking about propositional calculus such a statement could be $p\wedge\neg p$.

Now under any circumstances whatever, your two statements have the same truth value, namely, "false" - because they are always false.

If you consider the set consisting of these two statements, there is no way they can both be true simultaneously - because there is no way that even one of them can be true. So the set is inconsistent.

Answer 3

If you have two statements $A,B$ such that $A \Leftrightarrow B$, and have $A,B \Rightarrow (C \land \lnot C)$ you also have $A \Rightarrow (C \land \lnot C)$, so if $A$ is incosistent by itself, like $A=D \wedge \lnot D$ you are there.