Proofs About Subsets

There are many simple statements about sets which should be immediately obvious to students reading this text, but learning to write proofs for these types of statements will be instructive and useful in the proof writing discussed in the following chapters. Here are some of those simple statements that apply to all sets A, B, and C.

Some Statements About All Sets A, B, and C

• A  A [ B.

• A \ B  A.

• An.B [ C/  .A [ B/nC.

• .A [ B/ \ C  A [ .B \ C/.

• A [ B D B [ A, the Commutative Law of Union.

• A \ B D B \ A, the Commutative Law of Intersection.

• .A [ B/ [ C D A [ .B [ C/, the Associative Law of Union.

• .A\B/\C D A\.B\C/, the Associative Law of Intersection.

• A[ .B \ C/ D .A [ B/ \ .A [ C/, the Distributive Law of Union Over Intersection.

• A \.B [ C/ D .A \ B/ [ .A \ C/, the Distributive Law of Intersection Over Union.

• .A [ B/^cD A^c\ B^c, DeMorgan’s Laws.

• .A \ B/^cD A^c[ B^c, DeMorgan’s Laws.

The first four of these statements propose that one set is a subset of a second set.

From the definition of subset, for A  B to be true, it is required that for every x2 A, x must also be in B. There is a standard template for proofs of statements of this form:

TEMPLATE for proving A B for sets A and B

• SET THE CONTEXT: State what is being assumed about the sets A and B.

• ASSERT THE HYPOTHESIS: Let x 2 A.

• LIST IMPLICATIONS: Use the properties of set A to show x belongs to set B.

• STATE THE CONCLUSION: x 2 B. Therefore, by the definition of subset, A B.

For example, how would one prove the statement “For all sets A and B, A  A[ B?” Because this proof is supposed to apply to any sets A and B regardless of what properties they may possess, all that would be necessary for the “SET THE CONTEXT” part of the proof is a statement introducing to the reader the fact that the variables A and B will represent sets. Since A  A [ B exactly when every element of A is also an element of A [ B, the “ASSERT THE HYPOTHESIS” part of the proof needs to select a generic element of the set A so that the proof can conclude that the generic element is an element of set A [ B. The first two lines of the proof read:

Suppose that A and B are any two sets. Let x2 A.

The “LIST IMPLICATIONS” for this proof can be very short. It merely needs to show that the definition of “set union” implies that x is in the union A [ B. This completes the proof.

PROOF: A A [ B.

• SET THE CONTEXT: Suppose that A and B are any two sets.

• ASSERT THE HYPOTHESIS: Let x 2 A.

• LIST IMPLICATIONS: Since x 2 A, it is true that x 2 A or x 2 B.

• By the definition of set union x 2 A [ B.

• STATE THE CONCLUSION: Therefore, by the definition of subset, A A [ B.

Do the statements of this proof have to appear in exactly this order using exactly these words? Of course not. There can be many variations in what makes up a good proof. But it does not hurt to review why these statements make a good proof. The first line Suppose that A and B are any two sets just makes it clear to the reader that the variables A and B can be used to represent any two sets. Here is where the reader of the proof may well mentally choose two sets so that when reading the remainder of the proof, the reader can verify that the statements make sense when applied to those two sets. The second line Let x 2 A is required because by the definition of “subset,” one must show that each element of A is also an element of A[ B, so selecting an arbitrary element of A is the natural way to do this. The next line Since x 2 A, it is true that x 2 A or x 2 B is just a statement of logic that says if statement p is true, then statement p or q is also true. Of course, this particular p or q statement is exactly the definition of x being a member of A [ B, which is exactly what is needed to complete the proof.

2.3 Proofs About Sets 21 Could one have interchanged the third and fourth lines of this proof? Well, yes;

the proof would be complete if that were done, but the fact that the definition of set union is invoked right after its conditions are verified makes the statements of the proof flow smoothly. The reader facing the definition of set union in line three might wonder why that definition is being shown at that point. By placing that statement as the fourth statement where the proof reader has just seen that x 2 A or x 2 B, the proof reader will immediately see that the definition of set union applies. Note that each of the five statements in the proof has been placed on a separate line in the display box. This has been done merely to facilitate the discussion about that proof.

In practice, there is no requirement that these statements appear on a separate lines.

The second statement about all sets is A \ B  A. This can be proved using the same proof template as the first statement. Since this statement also applies to any two sets A and B, the first line of this proof will be the same as the first line of the previous proof. Because the assertion of the statement being proved is that A \ B is a subset of another set, the “ASSERT THE HYPOTHESIS” line of the proof would change to the assertion that x belongs to A \ B. After reading this second line, what does the proof reader know about x? Only that x belongs to the intersection of two sets. Thus, that only direction that the proof can proceed is to invoke the definition of set intersection to make the additional assertion that x 2 A and x 2 B. This is a statement of the form p and q, so logic allows the assertion that p is true, or, in this case, that x 2 A. This is the required “STATE THE CONCLUSION” statement, and the complete proof would be

PROOF: A\ B  A.

• SET THE CONTEXT: Suppose that A and B are any two sets.

• ASSERT THE HYPOTHESIS: Let x 2 A \ B.

• LIST IMPLICATIONS: By the definition of set intersection, x 2 A and x2 B.

• Thus, x 2 A.

• STATE THE CONCLUSION: Therefore, by the definition of subset, A\ B  A.

For a more substantial example, consider the third of the list of statements about sets An.B [ C/  .A [ B/nC. A proof of this statement will need to refer to the definition of set difference as well as the definitions of set union and subset. Since the statement being proved involves three sets, the “SET THE CONTEXT” part of the proof will need to refer to all three sets. The “ASSERT THE HYPOTHESIS”

statement will need to select an arbitrary element from An.B [ C/. To emphasize that the choice of which variable to use is arbitrary, this time use y rather than x to represent the arbitrarily chosen element. Once it is known that y 2 An.B [ C/, the only property of y that can be used is the fact that y is a member of a set difference. Thus, this would be a good time to invoke the definition of set difference.

That assures that y 2 A and y ….B [ C/. At that point one can use the definition of

set union to conclude that since y ….B [ C/ that y … B and y … C. Now these facts can be combined to get the “STATE THE CONCLUSION” statement required by the proof template. The complete proof would be

PROOF: An.B [ C/  .A [ B/nC.

• SET THE CONTEXT: Suppose that A, B, and C are any three sets.

• ASSERT THE HYPOTHESIS: Let y 2 An.B [ C/.

• LIST IMPLICATIONS: By the definition of set difference, y 2 A and y … .B [ C/.

• By the definition of set union y cannot be an element of either set B or set C, or it would be in B [ C.

• Also by the definition of set union, since y 2 A, y is also a member of A [ B.

• Now, y 2 .A [ B/ and y … C, so by the definition of set difference, y 2 .A [ B/nC.

• STATE THE CONCLUSION: Therefore, by the definition of subset, An.B [ C/  .A [ B/nC.

2.3.4 Exercises

Write proofs for each of the following statements.

1. For all sets A, B, and C,.A \ B/ \ C  A \ C.

2. For all sets A, B, and C,.A \ B/ \ .A \ C/  B \ C.

3. For all sets A, B, and C,.AnB/ \ .AnC/  An.B \ C/.