In logic, if p is true and q is true, what can be concluded about ~p and ~q?

In logic, if p is true and q is true, what can be concluded about ~p and ~q?

• A. Both are true
• B. Both are false
• C. Only ~p is true
• D. Only ~q is true

Answer: (B)

When we analyze the statements involving p and q, where both are given as true, we need to look at their negations, represented by ~p and ~q. Negation in logic implies the opposite of the original statement. Therefore, if p is true, its negation ~p is false. Similarly, if q is true, then ~q is also false. Since both propositions p and q are true, their negations, ~p and ~q, are both false. This leads us to conclude that both negated statements cannot be true at the same time, thus establishing that the correct conclusion is that both ~p and ~q are false. Understanding this relationship between a statement and its negation is fundamental in logic, and it confirms that when both p and q are true, their negations must be false.

When we analyze the statements involving p and q, where both are given as true, we need to look at their negations, represented by ~p and ~q.

Negation in logic implies the opposite of the original statement. Therefore, if p is true, its negation ~p is false. Similarly, if q is true, then ~q is also false.

Since both propositions p and q are true, their negations, ~p and ~q, are both false. This leads us to conclude that both negated statements cannot be true at the same time, thus establishing that the correct conclusion is that both ~p and ~q are false.

Understanding this relationship between a statement and its negation is fundamental in logic, and it confirms that when both p and q are true, their negations must be false.