Disjunction

Definition

Disjunction is a logical operation that combines two or more statements or conditions, where the result is true if at least one of the individual statements or conditions is true. It is a fundamental concept in the study of linear inequalities and absolute value inequalities.

5 Must Know Facts For Your Next Test

1. Disjunction is represented by the logical 'OR' operator, denoted by the symbol '∨'.
2. In a disjunctive statement, the truth of the overall statement is satisfied if at least one of the individual statements is true.
3. Disjunction is a key concept in solving linear inequalities and absolute value inequalities, as it allows for the representation of multiple solution sets.
4. Disjunctive inequalities can be used to model real-world situations where multiple conditions or constraints need to be considered.
5. The solution set for a disjunctive inequality is the union of the solution sets for the individual inequalities.

Review Questions

• Explain how disjunction is used in the context of linear inequalities.
  • In the context of linear inequalities, disjunction is used to represent situations where there are multiple possible solution sets. For example, a linear inequality like $x < 3 \vee x > 5$ has a solution set that includes all values of $x$ that are either less than 3 or greater than 5. The use of disjunction allows for the representation of these two distinct solution sets as a single compound inequality.
• Describe the relationship between disjunction and the union of solution sets.
  • The solution set for a disjunctive inequality is the union of the solution sets for the individual inequalities. This means that the final solution set includes all the values of the variable that satisfy at least one of the individual inequalities. Mathematically, if we have a disjunctive inequality like $x < 3 \vee x > 5$, the solution set is the union of the solution sets for $x < 3$ and $x > 5$, which would be $(-\infty, 3) \cup (5, \infty)$.
• Analyze the role of disjunction in solving absolute value inequalities.
  • Absolute value inequalities often involve disjunction to represent the two possible solution sets. For example, the inequality $|x - 2| < 4$ can be rewritten as a disjunctive inequality: $(x - 2 < 4) \vee (x - 2 > -4)$. This allows for the identification of the two solution sets, $(-2, 6)$ and $(0, 4)$, which together form the final solution set. The use of disjunction is crucial in solving absolute value inequalities, as it enables the representation of these distinct solution sets as a single compound inequality.

Related terms

Conjunction: A logical operation that combines two or more statements or conditions, where the result is true only if all the individual statements or conditions are true.

Negation: A logical operation that reverses the truth value of a statement, making a true statement false and a false statement true.

Compound Inequality: An inequality that involves more than one inequality, connected by the logical operations of disjunction or conjunction.