Consider the operators ◊ and □ defined by 𝑎 ◊ 𝑏 = 𝑎 + 2𝑏, 𝑎□𝑏 = 𝑎𝑏,…

Correct statements and why:

• Operator □ obeys the associative law: (a □ b) □ c = (ab)c = a(bc) = a □ (b □ c).

• Operator □ over the operator ◊ obeys the distributive law: a □ (b ◊ c) = a(b + 2c) = ab + 2ac, and (a □ b) ◊ (a □ c) = (ab) ◊ (ac) = ab + 2ac, so they are equal.

Why the other statements are false:

• Operator ◊ is not associative: (a ◊ b) ◊ c = a + 2b + 2c, while a ◊ (b ◊ c) = a + 2b + 4c, so they differ in general. Example: with a = b = c = 1, (1 ◊ 1) ◊ 1 = 5 but 1 ◊ (1 ◊ 1) = 7.

• ◊ does not distribute over □: a ◊ (b □ c) = a + 2(bc) whereas (a ◊ b) □ (a ◊ c) = (a + 2b)(a + 2c), which are not equal in general. Example: a = b = c = 1 gives 3 on the left and 9 on the right.

Final answer: The true statements are those asserting that multiplication (□) is associative and that multiplication distributes over ◊.