When is something logically equivalent

For instance, consider the 2 following statements:. If Sally wakes up late or if she misses the bus, she will be late for work. Therefore, if Sally arrives at work on time, she did not wake up late and did not miss the bus.

Logical analysis does not help determine the merit of an argument. Instead it helps to analyze the argument's form to determine if the truth of the conclusion follows from the truth of the preceding statements. While the content of the two above statements is different, their logical form is similar. If p or q, then r.

Therefore, if not r, then not p and not q. Compound Statements. Two statements are logically equivalent if, and only if, their resulting forms are logically equivalent when identical statement variables are used to represent component statements. Two statement forms are logically equivalent if, and only if, their resulting truth tables are identical for each variation of statement variables.

To test for logical equivalence of 2 statements, construct a truth table that includes every variable to be evaluated, and then check to see if the resulting truth values of the 2 statements are equivalent. I construct the truth table for and show that the formula is always true. The last column contains only T's.

Therefore, the formula is a tautology. Construct a truth table for. You can see that constructing truth tables for statements with lots of connectives or lots of simple statements is pretty tedious and error-prone. While there might be some applications of this e. The point here is to understand how the truth value of a complex statement depends on the truth values of its simple statements and its logical connectives.

In most work, mathematicians don't normally use statements which are very complicated from a logical point of view. Tell whether Q is true, false, or its truth value can't be determined. Since P is false, must be true. Hence, Q must be false. Since is false, is true. An "and" statement is true only when both parts are true.

In particular, must be true, so Q is false. I want to determine the truth value of. Therefore, the statement is true. You can't tell whether the statement "Ichabod Xerxes eats chocolate cupcakes" is true or false but it doesn't matter. If the "if" part of an "if-then" statement is false, then the "if-then" statement is true. Check the truth table for if you're not sure about this! So the given statement must be true. Two statements X and Y are logically equivalent if is a tautology.

Another way to say this is: For each assignment of truth values to the simple statements which make up X and Y, the statements X and Y have identical truth values. From a practical point of view, you can replace a statement in a proof by any logically equivalent statement. To test whether X and Y are logically equivalent, you could set up a truth table to test whether is a tautology that is, whether "has all T's in its column". However, it's easier to set up a table containing X and Y and then check whether the columns for X and for Y are the same.

Show that and are logically equivalent. Since the columns for and are identical, the two statements are logically equivalent. This tautology is called Conditional Disjunction. You can use this equivalence to replace a conditional by a disjunction. There are an infinite number of tautologies and logical equivalences; I've listed a few below; a more extensive list is given at the end of this section.

When a tautology has the form of a biconditional, the two statements which make up the biconditional are logically equivalent. Hence, you can replace one side with the other without changing the logical meaning. You will often need to negate a mathematical statement. To see how to do this, we'll begin by showing how to negate symbolic statements.

Write down the negation of the following statements, simplifying so that only simple statements are negated. I've given the names of the logical equivalences on the right so you can see which ones I used. I showed that and are logically equivalent in an earlier example. In the following examples, we'll negate statements written in words.

This is more typical of what you'll need to do in mathematics. The idea is to convert the word-statement to a symbolic statement, then use logical equivalences as we did in the last example. Use DeMorgan's Law to write the negation of the following statement, simplifying so that only simple statements are negated:.

Let C be the statement "Calvin is home" and let B be the statement "Bonzo is at the moves". This conditional statement is false since its hypothesis is true and its conclusion is false. Consequently, its negation must be true. Its negation is not a conditional statement. So, the negation can be written as follows:. This conjunction is true since each of the individual statements in the conjunction is true.

We have seen that it often possible to use a truth table to establish a logical equivalency. However, it is also possible to prove a logical equivalency using a sequence of previously established logical equivalencies. For example,. When proving theorems in mathematics, it is often important to be able to decide if two expressions are logically equivalent. Sometimes when we are attempting to prove a theorem, we may be unsuccessful in developing a proof for the original statement of the theorem.

However, in some cases, it is possible to prove an equivalent statement. Knowing that the statements are equivalent tells us that if we prove one, then we have also proven the other.

In fact, once we know the truth value of a statement, then we know the truth value of any other logically equivalent statement. This is illustrated in Progress Check 2. In Section 2. The logical equivalency in Progress Check 2. This gives us more information with which to work. Theorem 2. Conditional Statement. We have already established many of these equivalencies.