Write a formal negation for each of the following statements: a.
∀
fish x, x has gills. b.
∀
computers c, c has a CPU. c.
∃
a movie m such that m is over 6 hours long. d.
∃
a band b such that b has won at least 10 Grammy awards.

See below

Step-by-step explanation:

Let P(x) be a predicate over the variable x. Rules from logic tell us that the negation of ∀xP(x) is ∃x(¬P(x)). That is, the negation operator ¬ satisfies ¬(∀xP(x))≡∃x(¬P(x)). Similarly, ¬(∃xP(x))≡∀x(¬P(x)).

Apply these rules to each statement to obtain:

a) ∃ a fish x, x does not have gills.

b) ∃ a computer c, c does not have a CPU.

c) ∀ movie m, m is not over 6 hours long. ⇔  ∀ movie m, m is under 6 hours long.

d) ∀ band b, b has not won at least 10 Grammy awards. ⇔ ∀ band b, b has  won less than 9 Grammy awards.

  21.3

step-by-step explanation:

multiplying the given equation by c/sin(25°), we get

  c = 9/sin(25°) ≈ 21.2958 ≈ 21.3

we assume "c" is the length of ab, the side opposite angle c in triangle abc.

2nd one in the first row is correct