In this section, we will explore ways of combining statements to form new, more complex statements. A hallmark of our approach is that we will define the truth values of the statements completely formally (though generally the way you might expect). That is to say, the truth values of the combined statements will depend only on the truth values of the atomic statements of which they are composed. Deeper exploration will require a clear notion of logical equivalence, established via truth tables.
Suppose \(P\) is a statement. The negation of \(P\text{,}\) denoted \(\neg P\) and read “not \(P\)”, has the opposite truth value of \(P\) and is defined by the truth table found in Table 1.2.2.
We observe that Table 1.2.2 is our first encounter with a truth table. A truth table lists all possible truth values for a (compound) statement given all possible combinations of truth values of its atomic statements. Since there are only two possible truth values for the atomic statement \(P\text{,}\) there are only two rows in Table 1.2.2.
The conjunction of \(P\) and \(Q\text{,}\) denoted \(P \land Q\) and read “\(P\) and \(Q\)”, is true when both \(P\) and \(Q\) are true, and false otherwise. See Table 1.2.6.
The disjunction of \(P\) and \(Q\text{,}\) denoted \(P \lor Q\) and read “\(P\) or \(Q\)”, is true when \(P\) is true, \(Q\) is true, or both are true, and false otherwise. See Table 1.2.9.
Observe that the “or” connective defined in Definition 1.2.8 is distinct from the so-called “exclusive or” that is often used in, e.g., computer science. The statement \(P \lor Q\) is true so long as at least one of \(P,Q\) is true —or both!
The formalization of mathematical logic ramps up a bit when we consider conditional statements. It is important to remember that we define the truth value of the proposition \(P \Rightarrow Q\)formally based on the structure of the conditional statement and the truth values of the constituents \(P\) and \(Q\text{.}\) That is to say that there need not be a causal relationship between \(P\) and \(Q\text{!}\)
Suppose Dr. Janssen promises that, if everyone get an A in the class, then he will bring Defender sandwiches (on pretzel buns, with everything, as God intended) to celebrate on the last day of class 5
This is purely hypothetical.
. Unfortunately, a few students finish the course with an A –, so Dr. Janssen does not bring Defenders.
First, construct a truth table for \(P\Leftrightarrow Q\) based on Definition 1.2.18. Then, determine the truth values of the following statements. Identify which row(s) of your truth table you are using.
A fundamental skill in analyzing and proving mathematical statements is the ability to carefully describe their logical structure and, when appropriate, convert the statement into a logically equivalent statement. Ideally, the new equivalent form of the statement will have a structure more amenable to a particular type of proof or solution.
Choose at least two conditional statements we’ve mentioned in class and state the coverse, contrapositive, and inverse of each. Based on the statements you wrote, make a conjecture 7
A conjecture is a precise educated “guess” about what is true in general.
about how the truth values of these four implications (the original conditional statement, together with its converse, contrapositive, and inverse) seem to relate.
Use truth tables to identify any logical equivalences between \(P\Rightarrow Q\text{,}\)\(Q\Rightarrow P\text{,}\)\((\neg Q)\Rightarrow (\neg P)\text{,}\) and \((\neg P)\Rightarrow (\neg Q)\text{.}\)
The following pairs of logical equivalences, named for British mathematician Augustus De Morgan, allow us to convert conjunctions to disjunctions, and vice versa.
We will often interested in proving conditional statements in which our antecdents or consequents have multiple cases. The following equivalences will come in handy.
