27. For any matrices A and B for which the product AB is defined, the (i,j)-entry of AB equals a¡bj .

6. Negate the statement "Some cats are small but all roses are red."* a. No cat is small and some roses are not red. b. No cat is not small or some roses are red. c. No cat is small or some roses are not red. d. No cat is not and some roses are red.

Please show work, I've receive wrong responces that have been confusing me more than helping.

8. Rewrite the following argument as a conditional statement q

6. Fill in the blank with "all", "no", or “some" to make the following statements true. Note that "some" means one or more instances, but not all. • If your answer is "all", then give a brief explanation as to why. If your answer is "no", then give an example and a brief explanation as to why. • If your answer is "some", then give two specific examples that illustrate why your answer it not "all" or "no". Be sure to explain your two examples. An example must include either a graph or a specific function. (a) For (a, b). functions f, if f"(x) > 0 on the interval (a, b), then f'(x) < 0 on the interval (b) For functions f, if f(x) is a polynomial, then it is differentiable for all x. (c) For f(x) at exactly one point. functions f, the tangent line to f(x) at x = a will intersect the graph of In mathematics, we consider a statement to be false if we can find any examples where the statement is not true. We refer to these examples as counterexamples. Note that a counterexample is an example…

Determine if statement (3) follows from statements (1) and (2) by the Law of Detachment or the Law of Syllogism. If it does, state which law was used. If it does not, write INVALID

Translate each argument into symbolic form. Then determine whether the argument is valid or invalid. It is the case that x 29, but x ≥ 21, so x > 29. Choose the correct translation below. O A. pvq ~P_ .. q Choose the correct answer below. O A. The argument is valid. O B. The argument is invalid. O B. pvq ~q ..p C O C. pvq 9 .. ~p O D. pvq P ..-q