Logical Relationships: Truth Tables Bake the Cake

Logical Relationships: Truth Tables Bake the Cake

While truth tables might not obnoxiously materialize in the middle of the freeway as one drives to work, the world around us exists in an observable form easily described with truth tables. Every transformative scientist explores logical relationships in the development of any new idea. Our minds naturally compile information through conditional statements. In Discrete Mathematics at Maryville University, Dr. Dushanthi Herath (2022) explores logic as the conduit for discovering the validity of propositions: “Propositions are statements that can be either true or false but not both. In other words, propositional logic invents the rationality of the claims” (para. 1). Her presentation of propositions of logic demonstrates the relationship of negation, conjunction, disjunction, exclusive, conditional, biconditional, inverse, converse, and contrapositive statements and prepares active minds to explore complex combinations that set a foundation for examining the world mathematically. A riveting construction of (¬p ⊕ ¬q) ↔ (p ∧ q) encourages the critical analysis of truth tables using complex propositions:


Table 1

Truth Table for Complex Proposition (¬p ⊕ ¬q) ↔ (p ∧ q)


¬p

¬q

p

q

¬p ⊕ ¬q

p ∧ q

(¬p ⊕ ¬q) ↔ (p ∧ q)

F

F

T

T

F

T

F

F

T

T

F

T

F

F

T

F

F

T

T

F

F

T

T

F

F

F

F

T



Truth tables present a visual representation of conditional statements that indicate logical thinking and assist programmers in conquering complex relationships concretely. In programming, propositions are very similar to the conditional part of if-statements. In if-statements, additional operators symbolize the logic of these complex relationships. The complex proposition (¬p ⊕ ¬q) ↔ (p ∧ q) explored as if-statements in Java reveals how the use of propositions simplifies conditional statements and truth tables. This table illustrates the Java if-statement equivalent of the simple propositions that our complex propositions represent:


Table 2a

Comparable if-statements in Java for Complex Proposition (¬p ⊕ ¬q) ↔ (p ∧ q) Truth Table


¬p

¬q

p

q

if (!p) {

return true;

} else {

return false;

}

if (!q) {

return true;

} else {

return false;

}

if (p) {

return true;

} else {

return false;

}

if (q) {

return true;

} else {

return false;

}


Table 2b

Comparable if-statements in Java for Complex Proposition (¬p ⊕ ¬q) ↔ (p ∧ q) Truth Table


¬p ⊕ ¬q

p ∧ q

(¬p ⊕ ¬q) ↔ (p ∧ q)

if (!p && q || p && !q) {

return true;

} else {

return false;

}


if (p && q) {

return true;

} else {

return false;

}


if((!p && q || p && !q) == (p && q)){

return true;

} else {

return false;

}



Truth Tables provide a visual foundation for logical analysis. In a world full of logic and reason, how do you experience real-world situations illustrated through truth tables?


References

Herath, D. (2022). [Lecture notes MATH 311 course outline]. College of Arts and Sciences, Maryville University. https://maryville.instructure.com/courses/66399/pages/ negation-conjunction?module_item_id=5213807 

Rosen, K. (2018). Discrete Mathematics and Its Applications (8th ed.). McGraw Hill.


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