Logical Conjunction
In logic, a logical conjunction is a compound statement formed by combining two statements with the word 'and', symbolized as \(\land\). For the conjunction of two statements to be true, both of those statements must be true. If one or both are false, the conjunction is false. Imagine a scenario where you must wear both a hat and shoes to be considered fully dressed. If you're wearing just a hat or just shoes, or neither, you're not fully dressed. This is much like how conjunction works in logic.
For instance, if we have two propositions, A and B, the conjunction A \(\land\) B is only true when A is true and B is true. In the context of the exercise, the conjunction \(q \land r\) is false because while q is true, r is false. So, for the entire condition to hold, both parts must be true, but in this case, they're not.
Logical Disjunction
A logical disjunction is another type of compound statement that involves two separate statements joined by the word 'or', represented as \(\lor\). A disjunction is true if at least one of the statements is true. It's only false if both statements are false.
Example of Logical Disjunction
Suppose you can have an apple or an orange for a snack. Even if you pick just one, you still have a snack. The disjunction is satisfied with either choice, similar to how in logic, \(A \lor B\) requires either A or B (or both) to be true for it to be true. As seen in the exercise, the disjunction \(q \land r\) \(\lor r\) remains false because both components, \(q \land r\) and r, are false.
Negation
In logical terms, negation is akin to saying 'not'. It is the denial of a statement's truth. The negation of a true statement is false, and the negation of a false statement is true, symbolized as \(\sim\). In everyday language, it's like disagreeing with a claim.
Imagine someone says, 'It's raining.' If it's not raining, you're experiencing the negation of that statement. If we have a proposition A, its negation is written as \(\sim A\). In the exercise, the statement p is false, so the negation \(\sim p\) is true because it states 'not p'.
Logical Implication
This concept, logical implication, often denoted as \(\Rightarrow\), is a bit trickier. It's like saying if the first statement is true, then the second one is too. In formal logic, the implication A \(\rightarrow\) B is false only if A is true and B is false. In any other case, it's considered true.
Using a real-world scenario, think of it as a promise: If I wake up early (A), then I'll go for a run (B). The promise is broken (implication is false) only if I wake up early but don't go for a run. In our exercise, since the premise \(\sim p\) is true and the conclusion \(q \land r\) \(\lor r\) is false, the implication is false, and that's why the answer is (b) F.
