Proof Involving Sets Exploration

If k ∈ Z, then {n ∈ Z : n | k} ⊆ { n ∈ Z : n | k² }. Proof. Suppose k ∈ Z. We now need to show {n ∈ Z: n | k} ⊆ {n ∈ Z : n | k² }. Suppose a ∈ {n ∈ Z : n | k}. Then ii follows that a | k, so there is an integer c for which k = ac. Then k² = a2 c² . Therefore k² = a(ac² ), and from this the definition of (a)__________________________ gives a | k² . But a | k means that a ∈ { n ∈ Z : n | k² }. We have now shown {n ∈ Z : n | k} ⊆ {n∈ Z : n | k² }.

Use the methods introduced in this module to prove the following statements Suppose A,B and C are sets. Prove that if A ⊆ B, then A −C ⊆ B −C