Sets

4. Let A = {a, {a}, {{a}}} B = {ø, {a}, {a, {a}}} C = {a} Be subsets of S = {ø, a, {a}, {{a}}, {a, {a}}}. Find a) A C b) B C’ c) A B d) ø B e) (B C) A f) A’ B g) {ø} B 5. Let A = {x | x is the name of a former president of the US} B = {Adams, Hamilton, Jefferson, Grant} C = {x | x is the name of a state} Find: a) A B b) A C c) B C 6. Consider the following subsets of the set of all students: A = set of all computer science majors B = set of all physics majors C = set of all science majors D = set of all female students Using set operatio... click for more

Sets

8. Which of the following are true for all sets A, B, and C? a) B B = B b) (A’)’ = A c) (A – B) (B – A) = ø d) If A B = ø, then A is a proper subset of B e) B X A = A X B f) ø X A = ø g) ø {ø} = ø h) (A – B) (B – C) = A – C i) (A – C) (A – B) = A – (B C) 9. For any finite set S, let |S| denote the number of elements in S. If |A| = 3 and |B| = 4, find: a) |A X B| b) |A^2| c) |B^2| d) the maximum possible value for |A B| e) the minimum possible value for |A B| 10. Prove that A is a subset of (A B) where A and B are arbitrary sets. 11. Prove that the power set of A the power set of B is a subset of the power... click for more

Prove the following statements using induction:

This problem set has nine statements that must be proven using induction. Please see the attachment, because summation signs, product signs, etc. can't be understood in text. Some of the questions in this problem set are: 3. Let A be an infinite set, then show that |P(A)| = 2^|A| 8. Let n > 1 be an integer. Then, show that n can be written as a product of primes. See attached file for full problem description.

Questions about ordered pairs, ordered triples, binary/unary operations, postfix notation

1. Recall that ordered pairs must have the property that (x,y) = (u,v) if and only if x = u and y = v. a) Prove that {{x}, {x,y}} = {{u}, {u,v}} if and only if x = u and y = v. Therefore, although we know that (x,y) does not equal {x,y} , we can define the ordered pair (x,y) as the set {{x}, {x,y}}. b) Show by an example that we cannot define the ordered triple (x, y, z) as the set {{x}, {x,y}, {x,y,z}} 2. Which of the following are binary or unary operations on the given sets? For those that are not, where do they fail? a) x ◦ y = 1/x if x is positive S = set of all real numbers 1/(-x) if x is negative b) x ◦ y = xy (concatenation); S = se o... click for more

Matrices

Find a matrix A such that ... See attached file for full problem description.