Prove or disprove: For all integers a and b, if 𝑎𝑎2 ∤𝑏𝑏2, then 𝑎𝑎 ∤𝑏𝑏.

We say that an integer is waffy if it can be written in the form 3𝑘𝑘+ 1, where k is an integer.  Prove or disprove: The product of two waffy integers is also waffy.

2. Prove or disprove: For all integers a and b, if 𝑎𝑎2 ∤𝑏𝑏2, then 𝑎𝑎 ∤𝑏𝑏.

3. Prove or disprove: For all sets 𝐴𝐴,𝐵𝐵, and 𝐶𝐶, if 𝐴𝐴⊆𝐵𝐵, then 𝐴𝐴× 𝐶𝐶 ⊆𝐵𝐵× 𝐶𝐶.
(Recall that 𝐴𝐴× 𝐵𝐵 and 𝐴𝐴× 𝐶𝐶 are Cartesian products and think about what their elements
look like.)

4. Prove or disprove: For all sets 𝐴𝐴,𝐵𝐵, and 𝐶𝐶, 𝐴𝐴\(𝐵𝐵∪𝐶𝐶) = (𝐴𝐴\𝐵𝐵) ∪𝐶𝐶