3.3: Equivalence and Implication
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There are many ways to write the same logical equation. Too, various equations are implied by other equations or are contradicted. This section explores that idea. For instance, changing the terminology used to describe an idea, object, or field of study does not change those from what they are. An example are cloud providers who reinvent basic terminology in distributed systems so that their offering appears to be new and unique. An understanding of this area of thought will allow you to see through marketing hype and to simplify logical equations. Thus, you can bring clarity to your understanding and lower costs to systems described by logical expressions.
3.3 Equivalence and Implication
Consider two propositions generated by p and q: \(\neg (p \wedge q)\) and \(\neg p \vee \neg q\). At first glance, they are different propositions. In form, they are different, but they have the same meaning. One way to see this is to substitute actual propositions for p and q; such as p: I've been to Toronto; and q: I've been to Chicago.
Then \(\neg (p \wedge q)\) translates to "I haven't been to both Toronto and Chicago", while \(\neg p \vee \neg q\) is "I haven't been to Toronto or I haven't been to Chicago". Determine the truth values of these propositions. Naturally, they will be true for some people and false for others. What is important is that no matter what truth values they have, \(\neg (p \wedge q)\) and \(\neg p \vee \neg q\) will have the same truth value. The easiest way to see this is by examining the truth tables of these propositions.
In all four cases, \(\neg (p \wedge q)\) and \(\neg p \vee \neg q\) have the same truth value. Furthermore, when the biconditional operator is applied to them, the result is a value of true in all cases. A proposition such as this is called a tautology.
3.3.1 Tautologies and Contradictions
Definition 3.3.2: Tautology
An expression involving logical variables that is true in all cases is a tautology. The number 1 is used to symbolize a tautology.
Example 3.3.3: Some Tautologies
All of the following are tautologies because their truth tables consist of a column of 1's.
- \(\neg (p \wedge q)) \leftrightarrow (\neg p \vee \neg q)\)
- \(p \vee \neg p\)
- \((p \wedge q) \to p\)
- \(q \to (p \vee q)\)
- \((p \vee q) \leftrightarrow (q \vee p)\)
Definition 3.3.4: Contradiction
An expression involving logical variables that is false for all cases is called a contradiction. The number 0 is used to symbolize a contradiction.
Example 3.3.5: Some Contradictions
\(p \wedge \neg p\) and \((p \vee q) \wedge (\neg p) \wedge (\neg q)\) are contradictions.
3.3.2 Equivalence
Definition 3.3.6: Equivalence
Let S be a set of propositions and let r and s be propositions generated by S . r and s are equivalent if and only if \(r \leftrightarrow s\) is a tautology. The equivalence of r and s is denoted \(r \Leftrightarrow s\).
Equivalence is to logic as equality is to algebra. Just as there are many ways of writing an algebraic expression, the same logical meaning can be expressed in many different ways.
Example 3.3.7: Some Equivalences
The following are all equivalences:
- \((p \wedge q) \vee (\neg p \wedge q) \Leftrightarrow q\)
- \(p \to q \Leftrightarrow \neg q \to \neg p\)
- \(p \vee q \Leftrightarrow q \vee p\)
All tautologies are equivalent to one another.
Example 3.3.8: An equivalence to 1.
\(p \vee \neg p \Leftrightarrow 1\). All contradictions are equivalent to one another.
Example 3.3.9: An equivalence to 0.
\(p \wedge \neg p \Leftrightarrow 0\)
3.3.3 Implication
Consider the two propositions:
Table 3.3.10
x: The money is behind Door A; and
y: The money is behind Door A or Door B.
Imagine that you were told that there is a large sum of money behind one of two doors marked A and B, and that one of the two propositions x and y is true and the other is false. Which door would you choose? All that you need to realize is that if x is true, then y will also be true. Since we know that this can't be the case, y must be the true proposition and the money is behind Door B.
This is an example of a situation in which the truth of one proposition leads to the truth of another. Certainly, y can be true when x is false; but x can't be true when y is false. In this case, we say that x implies y.
Consider the truth table of \(p \to q\), Table 3.1.7. If p implies q, then the third case can be ruled out, since it is the case that makes a conditional proposition false.
Definition 3.3.11: Implication
Let S be a set of propositions and let r and s be propositions generated by S. We say that r implies s if \(r \to s\) is a tautology. We write \( r \Rightarrow s\) to indicate this implication.
Example 3.3.13: Disjunctive Addition
A commonly used implication called "disjunctive addition" is \(p \Rightarrow (p \vee q)\), which is verified by truth table Table 3.3.13.
If we let p represent "The money is behind Door A" and q represent "The money is behind Door B", \(p \Rightarrow (p \vee q)\) is a formalized version of the reasoning used in Example 3.3.12. A common name for this implication is disjunctive addition. In the next section we will consider some of the most commonly used implications and equivalences.
When we defined what we mean by a Proposition Generated by a Set, we didn't include the conditional and biconditional operators. This was because of the two equivalences \(p \to q \Leftrightarrow \neg p \vee q\) and \( p \leftrightarrow q \Leftrightarrow (p \wedge q) \vee (\neg p \wedge \neg q)\). Therefore, any proposition that includes the conditional or biconditional operators can be written in an equivalent way using only conjunction, disjunction, and negation. We could even dispense with disjunction since \(p \vee q\) is equivalent to a proposition that uses only conjunction and negation.
3.3.4 A Universal Operation
We close this section with a final logical operation, the Sheffer Stroke, that has the interesting property that all other logical operations can be created from it
Definition 3.3.14: The Sheffer Stroke
The Sheffer Stroke is the logical operator defined by the following truth table:
