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Determine whether each of the following compound propositions is satisfiable or unsatisfiable. Justify your answer. a. (p v ¬q) ^ (¬p v q) ^ (¬p V ¬q) b. (p V ¬q) ^ (¬p v q) ^ (q v r) ^ (q v ¬r) ^ (¬q v ¬r) c. (p → q) ^ (p → ¬q) ^ (¬p → q) ^ (¬p → ¬q) d. (-p → ¬q) ^(¬p → q)

Determine whether each of the following compound propositions is satisfiable or unsatisfiable. Justify your answer. a. (p v ¬q) ^ (¬p v q) ^ (¬p V ¬q) b. (p V ¬q) ^ (¬p v q) ^ (q v r) ^ (q v ¬r) ^ (¬q v ¬r) c. (p → q) ^ (p → ¬q) ^ (¬p → q) ^ (¬p → ¬q) d. (-p → ¬q) ^(¬p → q)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Additional Topics: Satisfiability A compound proposition is said to be satisfiable if there is an assignment of truth values to its variables that makes it true. For example, p Aq is true when p = T and q = T; thus, p^q is satisfiable. When no such assignment exists, the compound proposition is said to be unsatisfiable. For example, p^ ¬p is always false; thus, p A¬p is unsatisfiable. To show that a compound proposition is satisfiable, we need to find at least one assignment of truth values to its variables that makes it true. However, to show that a compound proposition is unsatisfiable, we need to show that every assignment of truth values to its variables makes it false. 6. Determine whether each of the following compound propositions is satisfiable or unsatisfiable. Justify your answer. a. (p v ¬q) ^ (¬p v q) ^ (¬p V ¬q) b. (p V ¬q) ^ (¬p v q) ^ (q v r) ^ (q v ¬r) ^ (¬q v ¬r) c. (p → q) ^ (p → ¬q) ^ (¬p → q) ^(¬p → ¬q) d. (-p → ¬q) ^ (¬p → q)
Transcribed Image Text:Additional Topics: Satisfiability A compound proposition is said to be satisfiable if there is an assignment of truth values to its variables that makes it true. For example, p Aq is true when p = T and q = T; thus, p^q is satisfiable. When no such assignment exists, the compound proposition is said to be unsatisfiable. For example, p^ ¬p is always false; thus, p A¬p is unsatisfiable. To show that a compound proposition is satisfiable, we need to find at least one assignment of truth values to its variables that makes it true. However, to show that a compound proposition is unsatisfiable, we need to show that every assignment of truth values to its variables makes it false. 6. Determine whether each of the following compound propositions is satisfiable or unsatisfiable. Justify your answer. a. (p v ¬q) ^ (¬p v q) ^ (¬p V ¬q) b. (p V ¬q) ^ (¬p v q) ^ (q v r) ^ (q v ¬r) ^ (¬q v ¬r) c. (p → q) ^ (p → ¬q) ^ (¬p → q) ^(¬p → ¬q) d. (-p → ¬q) ^ (¬p → q)
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