Chapter 2: Basic Structures: Sets, Functions, Sequences, Sums, and Matrices

Use the Schroder-Bernstein theorem to show that $\left(0,1\right)$and (0,1)have the same cardinality.

Compute each of these double sums

Suppose that g is a function from A to B and f is a function from B to C.

1. Show that if both f and g are one-to-one functions, then$\mathrm{f}\circ \mathrm{g}$is also one-to-one.
2. Show that if both f and g are onto functions, then $\mathrm{f}\circ \mathrm{g}$ is also onto.

Find the symmetric difference of the set of computer science majors at a school and the set of mathematics majors at this school.

Question: Suppose that g is a function from A to B and f is a function from B to C.

a) Show that if both f and g are one-to-one functions, then $\mathit{f}\mathbf{\circ }\mathit{g}$is also one-to-one. b) Show that if both f and g are onto functions, then $\mathit{f}\mathbf{\circ }\mathit{g}$is also onto.

We will establish distributive laws of the meet over the join operation in this exercise. Let A, B and C bezero-one matrices. Show that

a)$A\vee (B\wedge C)=(A\vee B)\wedge (A\vee C)$ b)$A\wedge (B\vee C)=(A\wedge B)\vee (A\wedge C)$

Show that the set S is a countable set if there is a function f from S to the positive integers such that$f^{-1}\left(j\right)$ is countable whenever j is a positive integer.

Let A be ann x nzero-one matrices. Let I be theidentity matrix. Show that $A\odot I=I\odot A=A$

Question: If f and $\mathit{f}\mathbf{\circ }\mathit{g}$are one-to-one, does it follow that g is one-to-one? Justify your answer.

Prove that 6 divides $n^3-n$ whenever n is a non negative integer.