Reference no: EM133266214

Consider the following claim:

Claim. {21 n : n ∈ Z} ∪ {14 n : n ∈ Z} ⊆ {7 n : n ∈ Z}.

a. Write the claim as an (equivalent) if-then statement.

b, Give a direct proof by cases that the claim is true. As a hint, you might want to prove the if-then statement that you constructed in (a).

To get full points you must use a mixture of formal notation and word explanations (e.g. the "column" format). Each step of your proof should have an explanation as to how/why you could make that a logical step. When in doubt, more detail is better than less.

c. State (but do not prove) the contrapositive of your statement from part (a).

d. State (but do not prove) the converse of your statement from part (a).

e. Give a disproof by counter-example of the converse from part (d). (That is, show that the converse is not true by providing an example that demonstrates it is not true.)

Remember that any disproof by counter-example not only provides the counter-example, but also an explanation as to why it is a counter-example.