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Chapter 3: Problem 87

Under which conditions is a conjunction true?

Short Answer

Expert verified
A conjunction is true only when both of its operands are true.

Step by step solution

01

Understanding Logical Conjunction

A logical conjunction is typically denoted as '鈭�'. It is an operation on two logical values, typically the values of two propositions, that produces a value that is true only if both of its operands are true. The conjunctive identity is '1', which is to say that AND-ing an expression with '1' will never change the value of the expression.
02

Truth Table for a Logical Conjunction Operation

A truth table for a conjunction operation can be defined as follows: If we consider 'A' and 'B' as two operands and '鈭�' is the conjunction operation, then: A 鈭� B = true, if both A and B are true; A 鈭� B = false, if A is true and B is false; A 鈭� B = false, if A is false and B is true; A 鈭� B = false, if both A and B are false.
03

Condition for Conjunction to be True

From the truth table described in Step 2, it is clear that a conjunction operation is true only when both operands are true. Therefore, the condition under which a conjunction is true is that both propositions/operands are true.

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Most popular questions from this chapter

Translate each argument into symbolic form. Then determine whether the argument is valid or invalid. It is the case that \(x<5\) or \(x>8\), but \(x \geq 5\), so \(x>8\).

Translate each argument into symbolic form. Then determine whether the argument is valid or invalid. You may use a truth table or, if applicable, compare the argument's symbolic form to a standard valid or invalid form. (You can ignore differences in past, present, and future tense.) If I'm tired, I'm edgy. If I'm edgy, I'm nasty. \(\therefore\) If I'm tired, I'm nasty.

Translate each argument into symbolic form. Then determine whether the argument is valid or invalid. You may use a truth table or, if applicable, compare the argument's symbolic form to a standard valid or invalid form. (You can ignore differences in past, present, and future tense.) If I'm at the beach, then I swim in the ocean. If I swim in the ocean, then I feel refreshed. \(\therefore\) If I'm not at the beach, then I don't feel refreshed.

Use a truth table to determine whether the symbolic form of the argument is valid or invalid. $$ \begin{aligned} &p \rightarrow q \\ &\frac{q \wedge r}{\therefore p \vee r} \end{aligned} $$

Determine whether each argument is valid or invalid. No \(A\) are \(B\), some \(A\) are \(C\), and all \(C\) are \(D\). Thus, some \(D\) are \(C\).
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