Let F(x, y) be the predicate that indicates if x and y are friends, with the domain of people. Which of the following statements in predicate logic expresses the statement that

"Every pair of friends has a friend in common"?

(only one correct answer)

a.∀x ∀y [ F(x, y) → ∀z ( (y ≠ z) ∧(x ≠ z) ∧ (F(x, z) ⋁ F(y, z)) ) ]
b.∀x ∀y [ F(x, y) → ∃z ( (y ≠ z) ∧ (x ≠ z) ∧ F(x, z) ∧ F(y, z) ) ]
c.∀x ∀y [ F(x, y) → ∃z ( (y ≠ z) ∧(x ≠ z) ∧ (F(x, z) ⋁ F(y, z)) ) ]

Write an equation in point-slope form for the line through the given point that has the given slope (-2,-7); m=-3/2

1- what do you think the product of a nonzero rational number and an irrational number is? is it rational or irrational? make use of variables, the closure property of integers, and possibly a proof by contradiction to prove your hypothesis.2- why do we have to specify that the rational number must be nonzero when we determine what the product of a nonzero rational number and an irrational number is? if the rational number were 0, would it give us the same result we found in the first question?

Simplify each expression for 18, 20, and 21