Understanding Conditional Statements and Proving the Negation | qp → q and Contradiction

qp→q_______∴-p

To answer this question, let’s break it down step by step

To answer this question, let’s break it down step by step.

The given statement is “qp → q”. This is an example of a conditional statement, also known as an implication. The general form of a conditional statement is “if p, then q”, where p represents the antecedent (the condition) and q represents the consequent (the result).

In this case, our antecedent is “qp” and our consequent is “q”. The symbol “→” denotes the implication or “if…then” relationship.

Now, to prove the statement “-p” (the negation of p), we can use a proof by contradiction. We assume that p is true and then try to derive a contradiction, which will imply that our assumption was false.

Let’s assume that p is true. From our original statement, “qp → q”, we can deduce that if qp is true, then q must also be true. However, we have assumed that p is true, which means qp is also true. Therefore, q must be true as well.

But we are trying to prove the negation of p, which is -p. If q is true, it contradicts the assumption that p is true, because p being true would imply that q is false. Therefore, we have arrived at a contradiction.

Since our assumption that p is true leads to a contradiction, we can conclude that p must be false, or -p is true.

Hence, we have proved that if the given statement “qp → q” is true, then -p must also be true.

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