Use the element method to prove that a set is empty, prove or disprove that a set is a subset of another set, or prove that two sets are equal.

Question

Let sets A, B, and C be defined as follows:

A = {x ∈ Z : x = 63r + 2 for some integer r}

B = {x ∈ Z : x = 7r + 2 for some integer r}

C = {x ∈ Z : x = 9r + 2 for some integer r} .

Prove or disprove the statement A ⊆ B ∪ C.

Answer 1

The statement is true.

Let sets A, B, and C be defined as follows:

A = {x ∈ Z : x = 63r + 2 for some integer r}

B = {x ∈ Z : x = 7r + 2 for some integer r}

C = {x ∈ Z : x = 9r + 2 for some integer r} .

A ⊆ B ∪ C.

Proof. Let a be an element of A. It suffices to prove the claim by showing that a is either in set B or in set C by the definition of the union of sets.

Then by definition of the set A, a=63r+2 for some integer r.

But 63r+2=7(9r)+2=7s+2 for some integer s=9r. s is an integer since 9 and r are both integers, integers are closed under multiplication, so 9r is also an integer.

Finally, since s is an integer, a=7s+2 is also an integer since integers are not only closed under multiplication but addition as well.

This shows that a=7s+2 is in B, and therefore B ∪ C too.

Q.E.D.

Remark. You could have also shown that a is in C instead, it doesn't matter. In order to show something is an element of a union of sets, you just have to show it's on one of those sets by showing it satisfies the properties of being in that set.