Question

Negate the following sentences. If \(f\) is a polynomial and its degree is greater than \(2,\) then \(f^{\prime}\) is not constant.

Short answer

The negation of the given statement is 'If \(f\) is not a polynomial or its degree is less than or equal to \(2,\) then \(f^{\prime}\) is constant.'

Step-by-step solution

Identify the hypothesis and the conclusion

The hypothesis of the given sentence is 'if \(f\) is a polynomial and its degree is greater than \(2\)', while the conclusion is 'then \(f^{\prime}\) is not constant.'

Negate the hypothesis

The negation of the hypothesis 'if \(f\) is a polynomial and its degree is greater than \(2\)' is 'if \(f\) is not a polynomial or its degree is less than or equal to \(2\)'. This negation has been achieved using De Morgan's Laws where the antithesis of a conjunction ('and') is the disjunction ('or') of the negations.

Negate the conclusion

The negation of the conclusion 'then \(f^{\prime}\) is not constant' is 'then \(f^{\prime}\) is constant'. This is simply reversing the statement.

Construct the negation of the initial statement

The final step is to piece together the negated hypothesis and negated conclusion from the above steps. The result should be 'If \(f\) is not a polynomial or its degree is less than or equal to \(2,\) then \(f^{\prime}\) is constant.'

Key concepts

Logic in Mathematics

Logic forms the backbone of mathematics, providing a framework for constructing and evaluating arguments based on rigorous rules. In the context of negating statements, logic helps us understand how to properly invert the meaning of a proposition without altering its truth value.

When working with logical statements in mathematics, we generally encounter two parts: the hypothesis (the 'if' part) and the conclusion (the 'then' part). Negating a statement involves switching the hypothesis and conclusion from affirming something, to denying it. However, it’s important to do this correctly, as improper negation can lead to a different meaning than intended.

As seen in the exercise solution, negating a complex hypothesis that involves conjunctions ('and') requires careful application of logical principles. Students should pay special attention to how changing 'and' to 'or', and the implications of such transformation, can affect the overall negation of a statement.

Polynomial Functions

Polynomial functions are algebraic expressions that involve sums of powers of a variable. They come in various degrees, which correspond to the highest power of the variable in the function. Understanding the degree of a polynomial is essential when analyzing its properties and behavior.

The given exercise touches on polynomials of degree greater than 2. Such functions are interesting because their derivatives, which represent the slopes of their tangent lines, are themselves polynomials with one degree less. This means that for any polynomial with a degree greater than 2, its derivative will not be a constant function. On the other hand, if the degree is 2 or less, the derivative may very well be constant.

When students engage with polynomial functions, they should remember the relationship between the degree of a polynomial and the behavior of its derivative. This concept is often tested in calculus and is a fundamental property of polynomials.

De Morgan's Laws

De Morgan's Laws are crucial rules of logic that dictate how to relate the negation of conjunctions and disjunctions. These laws state that the negation of a conjunction ('and') is logically equivalent to the disjunction ('or') of the negations, and vice versa.

In the exercise, De Morgan's Laws come into play in Step 2, where the original hypothesis includes an 'and' statement. To negate it, we don't simply negate each part but also switch 'and' to 'or'. This is the essence of De Morgan's Laws. Applying these laws can sometimes seem counterintuitive, which is why it is essential to grasp their logic to avoid common mistakes.

Example of De Morgan's Laws

• Original statement: It is raining and cold (A and B).
• Negated using De Morgan's: It is not raining or it is not cold (not A or not B).

As students progress in mathematics, recognizing and applying De Morgan's Laws becomes routine especially in subjects involving logic, set theory, and computer science.