Problem 6
Area codes are used to distinguish phone numbers for which the last seven digits are the same. If you have 35,000,000 phone numbers in a state and an area code can distinguish approximately 900,000 phone numbers, how many area codes are needed to distinguish the phone numbers of this state?
Problem 6
Let \(X=\\{0,1,2\\} \subseteq \mathbb{R}\). List all eight strictly increasing sequences of elements of \(X\). The ordering is \(<\) on \(\mathbb{R}\). List all subsequences of the sequence \(x, y, z\).
Problem 7
Let \(A=\\{1,2,3,4\\}\). Let the functions \(F, G,\) and \(H\) be given with domain and codomain \(A\) defined as \(F(1)=3, F(2)=2, F(3)=2,\) and \(F(4)=4\) \(G(1)=1, G(2)=3, G(3)=4,\) and \(G(4)=2\) \(H(1)=2, H(2)=4, H(3)=1,\) and \(H(4)=3\) Find the following: (a) \(F \circ G\) (b) \(H \circ F\) (c) \(G \circ H\) (d) \(F \circ G \circ H\)
Problem 7
There are 35.000 students at State University, Each student takes four different courses each term. State University offers 999 courses each term. The largest classroom on campus holds 135 students. Is this a problem? If so, what is the problem?
Problem 7
Prove that the function \(F:(0,1) \rightarrow \mathbb{R}\) defined as \(F(x)=(1 / 2-x) /(x(1-x))\) is a biiection.
Problem 8
Let \(X=\\{a, b\\}\) (a) There are nine partial functions \(F: X \rightarrow X\). List them. (b) There are four functions \(F: X \rightarrow X\). List them. (c) List all \(l-l\) functions \(F: X \rightarrow X\). (d) List all onto functions \(F: X \rightarrow X\).
Problem 8
At Bridgetown University, there are 45 time periods during the week for scheduling classes, Use the Generalized Pigeon-Hole Principle to determine how many rooms (at least) are needed if 780 different classes are to be scheduled in the 45 time slots.
Problem 9
For sets \(X, Y,\) and \(Z\), let \(F: X \rightarrow Y\) and \(G: Y \rightarrow Z\) be \(I-I\) correspondences. Prove that \((G \circ F)^{-1}=F^{-1} \circ G^{-1}\)
Problem 9
Let \(X=\\{-1,0,1,2\\}\) and \(Y=\\{-4,-2,0,2\\},\) Detine the function \(F: X \rightarrow Y\) as \(F(x)=x^{2}-x\). Prove that \(F\) is neither \(1-1\) nor onto.
Problem 9
Suppose someone (say, Aesop) is marking days in some leap year (say, 2948). You do not know which days he marks, only how many. Use this to answer the following questions. (Warning: Some, but not all, of these questions use the Pigeon-Hole Principle.) (a) How many days would Aesop have to mark before you can conclude that he marked two days in January? (b) How many days would Aesop have to mark before you can conclude that he marked two days in February? (c) How many days would Aesop have to mark before you can conclude that he marked two days in the same month? (d) How many days would Aesop have to mark before you can conclude that he marked three days in the same month? (e) How many days would Aesop have to mark before you can conclude that he marked three days with the same date (for example, the third of three different months, or the 3 ist of three different months)? (f) How many days would Aesop have to mark before you can conclude that he marked two consecutive days (for example, January 31 and February 1 )? (g) How many days would Aesop have to mark before you can conclude that he marked three consecutive days?
