Section 1.2: Propositional Equivalences In the process of reasoning, we often replace a known statement with an equivalent statement that more closely addresses the result that we are trying to establish. To ensure that this process produces a valid mathematical argument, we must be able to verify that the statement we are replacing does indeed have the very same meaning as the one we replace it with. Two such statements are said to be logically equivalent.

Tautologies and Contradictions Def: A compound proposition that is always true, regardless of the assignment of truth values to its component propositions, is called a tautology. A compound proposition that is always false is called a contradiction. A proposition that is neither a tautology nor a contradiction is called a contingency. Ex: 1.“p   p” is a tautology.[All T’s in the Truth Table.] 2.“p  p” is a contradiction. [All F’s in the Truth Table.] 3.“p   p” is a contingency. [At least one of each in TT.]

Logical Equivalences Def: Two compound propositions are called logically equivalent if they have the same truth value for every possible truth value assignment to the component propositions. Ex: 1.“p  q” is logically equivalent to “  q   p” (as we have seen). [Truth Table] [What is the relationship?] 2.“  (p  q)” is logically equivalent to “  p  q”. [T Table]

Note that if two propositions (which we label as p and q) are logically equivalent, then the biconditional of the two propositions (p  q) is a tautology [since it is true when p and q have the same truth value, so it is always true if p and q have the same truth value in all cases]. For this reason, we can denote that two propositions p and q are logically equivalent by stating that p  q is true. [What it means for a compound proposition to be true is that the proposition is a tautology, true in all cases]. Ex: “p  q” is logically equivalent to “  p  q”. [Why w/out TT?] Recall that an implication is true whenever the hypothesis is false (  p) or when the consequence is true (q), or both. It is only false when both the hypothesis is true and the consequence is false. So what we have determined is that “p  q   p  q” is true.

EquivalenceName p  T  p; p  F  p Identity Laws p  F  F; p  T  T Domination Laws p  p  p; p  p  p Idempotent Laws  (  p)  p Double Negation Law p  q  q  p; p  q  q  p Commutative Laws (p  q)  r  p  (q  r) (p  q)  r  p  (q  r) Associative Laws [no need for parens] p  (q  r)  (p  q)  (p  r) p  (q  r)  (p  q)  (p  r) Distributive Laws  (p  q)   p   q  (p  q)   p   q De Morgan’s Laws [generalizes] p  (p  q)  p; p  (p  q)  p Absorption Laws p   p  T; p   p  F Negation Laws

Logical Equivalences Involving  p  q   p  q p  q   q   p p  q   p  q p  q   (p   q)[Transform]  (p  q)  p   q (p  q)  (p  r)  p  (q  r) (p  r)  (q  r)  (p  q)  r (p  q)  (p  r)  p  (q  r) (p  r)  (q  r)  (p  q)  r

Duals If we have a proposition involving only propositional variables (p, q, r, s, t, etc.), negation(  ), disjunction(  ), conjunction(  ), and the constants T and F, then we can form what is called the dual of the proposition. The dual of the proposition is formed by replacing: 1. Each T with F and each F with T 2. Each  with  and each  with  This results in a new proposition which is called the dual of the original proposition. Ex: The dual of “(p  q)  (q  p)” is “(p  q)  (q  p)”. The dual of “p  T” is “p  F”.

The dual is important because of the following property. If we have two logically equivalent compound propositions (say p and q) then the duals of the two propositions are logically equivalent as well. That is, if p  q then p*  q*. Ex: We know that p  T  T. So p  F  F. In fact, if we look back at our table of logical equivalences, we will find that (with the exception of double negation) they come in pairs. And if we take one of the equivalences from a pair, we find that it is the dual of the other equivalence from the pair. The example above illustrates this for the domination laws. Taking the dual of both sides of a logical equivalence allows us to generate another logical equivalence. This is a useful technique. Aside: A truth table should be constructed for a compound proposition, not simply filled in.

Summary We have seen a great deal of logical equivalences. A very basic logical equivalence can be established by using a truth table to see if the two compound propositions have the same truth value in all cases. This becomes very tedious, especially when we are dealing with compound propositions that involve a large number of constituent propositions. In general, to construct a truth table for a compound proposition involving n constituent propositions, we must consider 2 n possibilities (rows of the truth table). Another method that can be used to establish a logical equivalence is to start with one compound proposition and then selectively replace parts of the proposition with logically equivalent pieces by using known logical equivalences. This becomes the method of choice when we do deal with large propositions. It is also preferable because by executing these transformations, we often understand the reason for logical equivalence much better than we do by constructing a TT.

Homework problems from Section 1.2 Problems 6, 8, 10, 12, 15, 21, and 30 from this section will be included on the next homework assignment.