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Chapter 3: Problem 10

Construct a truth table for the given statement. \(p \rightarrow(q \vee r)\)

Short Answer

Expert verified
The truth table for the statement \(p \rightarrow (q \vee r)\) is as follows: | p | q | r | q \(\vee\) r | p \(\rightarrow\) (q \(\vee\) r) ||---|---|---|---------|-------------------|| T | T | T | T | T || T | T | F | T | T || T | F | T | T | T || T | F | F | F | F || F | T | T | T | T || F | T | F | T | T || F | F | T | T | T || F | F | F | F | T |

Step by step solution

01

Understand the Variables

Set up the possible combinations of truth values for the variables p, q, and r. Since there are three variables, there will be \(2^3 = 8\) combinations, ranging from true (T), true (T), true (T) to false (F), false (F), false (F).
02

Determine the Truth Value of the Disjunction

Calculate the truth value of the inner disjunction (\(q \vee r\)) for each row in the table. In a disjunction, the resulting statement is true if at least one of the elements is true.
03

Determine the Truth Value of the Implication

Calculate the truth value of the entire implication statement (\(p \rightarrow (q \vee r)\)) for each row in the table. In an implication, the statement is only false when the first statement is true and the second statement is false.

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