Which of the following statements are true and which are false. justify your answers. hint: this is a great problem. you need to be particularly careful about the order of the quantifiers (∀ and ∃).
(a) ∃ ∈ ℝ such that ∀ ∈ ℝ 2 + 2 = 9.
(b) ∀ ∈ ℝ, ∃ ∈ ℝ such that 2 < +1.
(c) ∀ ∈ ℤ +, ∃ ∈ ℤ + such that = + 1.
(d) ∀ ∈ ℤ, ∃ ∈ ℤ such that = + 1.
(e) ∃ ∈ ℝ such that ∀ ∈ ℝ, = + 1.
(f) ∀ ∈ ℝ +, ∃ ∈ ℝ + such that = 1.
(g) ∀ ∈ ℝ, ∃ ∈ ℝ such that = 1.
(h) ∀ ∈ ℤ + and ∀ ∈ ℤ +, ∃ ∈ ℤ + such that = −
(i) ∀ ∈ ℤ and ∀ ∈ ℤ, ∃ ∈ ℤ such that = −
(j) ∃ ∈ ℝ + such that ∀ ∈ ℝ +, <
(k) ∀ ∈ ℝ +, ∃ ∈ ℝ + such that <

As a project for math class, two students devised a game in which 3 black marbles and 2 red marbles are put into a bag. first the players must decide who is playing black marbles and who is playing red marbles. then each player takes a turn at drawing a marble, noting the color, replacing the marble in the bag, and then drawing a second marble and noting the color before returning it to the bag. the point scheme for the game is detailed in the table below. point values for marble game black marble points red marble points both black: +2 points both red: +4 points different colors: –1 point different colors: –1 point both red: 0 points both black: 0 points