Assignment 2: Basic Set Operations
Due: December 12, 2019
Make sure to submit your homework on SINGLE-SIDED 8.5\(\times \)11 INCH PAPER. Write your NAME ON EVERY PAGE and DO NOT STAPLE OR FOLD the sheets.
In this and future homeworks, questions marked as optional will not be used for quizzes or final exam questions.
- 1.
- Find a counterexample to the statement, “All odd natural numbers are prime.” What is the least counterexample (NOTE: 1 is NOT prime)?
- 2.
Here are some standard definitions of set operations. I’m using the symbol “\(:=\)” to mean “equals by definition.” For any sets \(A\) and \(B\),
- Union:
- \(A\cup B := \{ x \st x\in A \myor x\in B \}\).
- Intersection:
- \(A\cap B := \{ x \st x\in A \myand x\in B \}\).
- Difference (Relative Complement):
- \(A-B := \{ x \st x\in A \myand x\notin B \}\).
- Symmetric Difference:
- \(A\symdiff B = (A-B)\cup (B-A)\).
For each of the four operations above, redraw the Venn diagram below, shading the region corresponding to the result of the operation.
- 3.
- Illustrate via a venn diagram the following identity for any sets \(A\) and \(B\). \[ A\symdiff B = (A\cup B) - (A\cap B). \]
- 4.
- Prove the identity in the last problem directly for any sets \(A\) and \(B\).
- 5.
- Let \(A\) and \(B\) be any sets. Which of the following sets must be empty? For each of those that need not be, give a specific, concrete example where it is not (your examples should be as small as possible).
- (a)
- \(A\cup \emptyset \)
- (b)
- \(A\cap \emptyset \)
- (c)
- \(A-\emptyset \)
- (d)
- \(A\times \emptyset \)
- (e)
- \(A-B\)
- (f)
- \(A-A\)
- (g)
- \((A-B)\cap (B-A)\)
- (h)
- \((A-B)-(B-A)\)
- (i)
- \(A-(B-A)\)
- (j)
- \(A-(A-B)\)
- (k)
- \(A-(A-A)\)
- (l)
- \((A-B)\cap B\)