Proof Involving Sets Exploration
Fill in the blank of the following proof.
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 }.
Your Turn.
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