Question

If both \(\mathrm{p}\) and \(\mathrm{q}\) are true (a) \(\mathrm{p} \Lambda \mathrm{q}\) is true (b) \(\mathrm{p} \mathrm{V} \mathrm{q}\) is false (c) \(\mathrm{p} \Rightarrow \mathrm{q}\) is false (d) None of them

Short answer

The correct statement is (a) \(p \Lambda q\) is true, as both p and q are true (T) and the evaluated expression using the AND operator is also true (T).

Step-by-step solution

Evaluate p AND q (p Λ q)

Recall the truth table for the AND operator: p | q | p Λ q T | T | T T | F | F F | T | F F | F | F Since both p and q are true (T), the expression p AND q (p Λ q) is also true (T). Therefore, the statement (a) is correct.

Evaluate p OR q (p V q)

Recall the truth table for the OR operator: p | q | p V q T | T | T T | F | T F | T | T F | F | F Since both p and q are true (T), the expression p OR q (p V q) is also true (T). Therefore, the statement (b) is incorrect.

Evaluate p IMPLIES q (p => q)

Recall the truth table for the IMPLIES operator: p | q | p => q T | T | T T | F | F F | T | T F | F | T Since both p and q are true (T), the expression p IMPLIES q (p => q) is also true (T). Therefore, the statement (c) is incorrect.

Conclusion

Based on the analysis of the logical expressions, the only correct statement is (a) p Λ q is true. Therefore, the answer is (a).

Key concepts

Truth Tables

Understanding truth tables is essential for evaluating logical statements in mathematics. A truth table is a handy tool that lists all possible truth values of logical variables or expressions. Each row in the table represents a possible combination of true (T) or false (F) values for the variables involved in the logical statements.

A truth table allows us to see the result of applying logical operators like AND, OR, and IMPLIES to these values. This is crucial because it provides a systematic method to determine the truthfulness of complex statements. By laying out all the scenarios, students can easily assess the validity of the logical operations they are performing.

Logical Conjunction

The term logical conjunction refers to the logical AND operator, often symbolized by \(\Lambda\), \(\cdot\), or \(\&\). It combines two statements and returns true only if both statements are true. In our example, when we evaluate \(p \Lambda q\), we check the conjunctive relationship between \(p\) and \(q\).

Here is a simplified breakdown of this operator's truth table: If \(p\) and \(q\) are both true, then \(p \Lambda q\) is true. This operator is fundamental in building more complex logical statements, and understanding it is the cornerstone of logical reasoning in mathematics.

Logical Disjunction

Conversely, the logical disjunction is about the logical OR operator, denoted by \(\mathrm{V}\). This operator considers two statements and will return true if at least one of the statements is true. It's an inclusive 'or,' which means it also returns true when both statements are true.

In our exercise, we consider \(p \mathrm{V} q\), and since either \(p\) or \(q\) or both are true, the resulting value of this disjunction is true. It's the perfect operator to use when a statement can be validated by multiple conditions.

Logical Implication

The logical implication, or the IMPLIES operator, represented by \(\Rightarrow\), tells us about a condition relationship. It states that if the first statement (the antecedent) is true, then the second statement (the consequent) must also be true for the entire implication to hold true. The only scenario where the implication fails is when a true antecedent leads to a false consequent.

In the context of our problem, \(p \Rightarrow q\) evaluates to true because both \(p\) and \(q\) are indeed true. Logical implication is essential in mathematical proofs where the validity of one statement depends on the truth of another.