Final answer:
Discrete math logical proof involves understanding statements, negations, conditionals, and counterexamples. A statement's negation is the opposite of the initial statement. The law of noncontradiction implies the law of the excluded middle, stating that for any proposition, either it or its negation is true.
Step-by-step explanation:
When discussing discrete math logical proof, it's important to understand various components of logic, such as statements, negations, conditional statements, and counterexamples.
To offer an example of a statement and its negation: If 'P' is 'It is raining,' then the negation of 'P' would be 'It is not raining.' These are simple, clear opposites of one another.
Concerning the law of noncontradiction and how it implies the law of the excluded middle: The law of noncontradiction states that a proposition cannot both be true and not true at the same time. From this principle, it logically follows that for any particular proposition, either that proposition is true, or its negation is true (law of the excluded middle), since there is no middle ground where the proposition can be both true and false.
For a conditional statement example: 'If it is snowing, then school will be closed.' Here, 'it is snowing' is the sufficient condition, and 'school will be closed' is the necessary condition. If the sufficient condition is met, the necessary condition follows.
A counterexample is a scenario which disproves a statement or argument by showing an instance where the premises are true, but the conclusion is false. For instance, consider the argument 'If a person is a teacher, then they work at a school.' A counterexample might be a private tutor who is a teacher but does not work at a school.