Question

Match the statement to be proved with the correct use of proof by cases.

Answer 1

The use of counterexamples in proof by cases is to demonstrate situations where premises are true, but the conclusion is false, thereby proving an argument invalid. Disjunctive syllogism is an example of where counterexamples can be applied to test the soundness of reasoning.

Step-by-step explanation:

In the context of proof by cases, the statement to be proved would be matched with scenarios where the premises of an argument are true, yet the conclusion is still false, thus identifying a counterexample. For instance, if we are examining a disjunctive syllogism, we would need to find a case where both premises are true but the conclusion is invalid. A counterexample works as an evidence against the universal validity of the argument, showing that at least in one situation the reasoning does not hold up.

As an example, take the statement 'If it rains, I will get wet or I will use an umbrella.' A counterexample could be a situation where it rains, and I neither get wet nor use an umbrella because I could be indoors. This indicates the argument is invalid because the conclusion (getting wet or using an umbrella) is not necessarily true even when the premise (it rains) is true. Thus, counterexamples are crucial in proving that a deductive argument is invalid.

Answer 2

Proof by cases involves demonstrating a statement holds true in various scenarios, and counterexamples in deductive reasoning show an argument's invalidity by revealing premises that are true yet conclude falsely. Articulating the significance of a proof relates it to a broader context or thesis. Topic sentences in an argument can serve to reason, illustrate, explain or provide evidence for the claim.

Step-by-step explanation:

To match the statement to be proved with the correct use of proof by cases, it's important to understand how counterexamples and deductive inferences work. When attempting to prove a statement with a proof by cases, you are effectively showing that the statement holds true in multiple distinct scenarios or instances, covering all possible options.

In the context of deductive reasoning, if an argument is invalid, a counterexample can be used to demonstrate this invalidity by providing a situation where the premises hold true but the conclusion does not follow. This counters the notion of a disjunctive syllogism, which states that if one statement in an 'or' construction is false, the other must be true, assuming the other aspects of the reasoning are sound.

For example, consider the assertion 'If it rains, the ground will be wet, or I will use an umbrella.' The counterexample might be 'It rained, the ground is dry because the sun dried it quickly, and I did not use an umbrella.' Here, the premises are true, but the conclusion is false, revealing the flaw in the argument structure.

An argument that articulates why what you've just proven matters is typically one that links the proof to a broader thesis or body of knowledge, demonstrating the relevance and impact of the proof.

When it comes to statements such as topic sentences, these can serve different roles including: a reason for the topic sentence's claim; an illustration of the topic's point; an explanation of the point; or evidence that demonstrates the topic sentence's point.

The complete question is: match the statement to be proved with the correct use of proof by cases. is: