Two #202204281244 are logical equivalence when they share the same truth values# for all situation. For example: \(p\) and \(\sim (\sim p)\) are logical equivalence. The following is their truth table:
| \(p\) | \(\sim p\) | \(\sim (\sim p)\) |
|---|---|---|
| T | F | T |
| F | T | F |
Their relationship can be denoted \(p \equiv \sim (\sim p)\). More logically equivalence examples in 202205061240#.
The negation of a Universal Statement is logically equivalent# to an 202204281254. Formally, this means that \(\sim (\forall x \in D, Q(x)) \equiv \exists x \in D \text{ such that } \sim Q(x)\). For instance, the negation of the statement “All books have a cover” is “There is at least one book which doesn’t have a cover”. It is tempting to say that the negation should be “No books have a cover”, but it is enough to disprove the above statement by just one counterexample.
All following laws are #logically equivalence. t denotes tautology (always true) and c denotes contradiction (always false). \(\equiv\) indicate that the left-hand side is identical to the right-hand side.
Note the similarity with the contrapositive of the 202205062055. Therefore, we can know that Modus Tollens is logically equivalent to 202206101747.
The negation of an Existential Statement is logically equivalent# to a 202204281245. Formally, this means that \(\sim (\exists x \in D \text{ such that } Q(x)) \equiv \forall x \in D, \sim Q(x)\). For instance, the negation of the statement “Some books have a cover” is “All books don’t have a cover” or “No books have a cover”. As you have noticed, the negation of it is to disprove every element in the set has the property \(Q(x)\), which is quite similar to the reasoning of the method of exhaustion.
Two digital logic circuits can be said are the same if they have the same input/output table. This can be proved by their corresponding 202206292207# to see whether they are logically equivalent#.
Denying The Antecedent or Inverse Error is a #202206172053 where one draw their conclusion from premises by replacing the premise \(p \rightarrow q\) by its inverse. From 202205062055# we know that the inverse of the conditional statement is not logically equivalence# to itself. The following shows its form:
The contrapositive of a conditional statement is logically equivalence# to the conditional statement itself. Both converse and inverse form of the conditional statement is not logically equivalence to it (the conditional statement), but they are logically equivalence to themselves (converse and inverse).
There is another form of Conditional Statement which is more restrictive, the Biconditional. It is denoted \(p \leftrightarrow q\) which means \(p\) if and only if \(q\). For example: Human is mammal if, and only if primate is mammal. We can either say that “If human is mammal, then primate is mammal” or “If primate is not mammal, then human is not mammal”. Sometimes, the phrase “if and only if” will be abbreviated to “iff”. Note that the statement \(P\) and \(Q\) are logically equivalence# if and only if \(P \leftrightarrow Q\) is a tautology.
Affirming The Consequent or Converse Error is a kind of #202206172053 where one draw a conclusion from the premises as if it is conversed. As we can see in 202205062055#, the converse of the conditional statement is not logically equivalence# to itself. The form could be seen below: