1) For A = {a, b, c, d, e} and B = {yellow, orange, blue, green, white, red, black}.
a) Define a relation R from A to B that is a function and contains at least 4 ordered pairs. Answer: (a,yellow),(b,yellow),(c,blue),(d,green),(e,red).
b) What is the domain of this function? Answer: all of A
c) What is the range of this function? Answer: Range = values={yellow,blue,green,red}

2) Define functions f: â„ â†’ â„ and g: â„ â†’ â„ by f(a) = 2 + a and g(b) = 3b â€“ 1. Find the following, showing the steps to get to your solution.
a) (f â—‹ g) (0). f(g(0))=f(3*0-1)=f(-1)=2+-1=1
b) (g â—‹ f) (1). g(f(1))=g(2+1)=g(3)=3*3-1=8.
c) (f â—‹ g) (x). f(g(x))=f(3x-1)=2+3x-1=3x+1.
d) (g â—‹ f) (x). g(f(x))=g(2+x)=3(2+x)-1=5+3x

3) Let A = {CA, NH, IL, OH, SC, WV, PA, TX} and B = {book, table, chair, fork, road, car}. Using at least 5 ordered pairs, specify the following:
a) Define a function f from A to B that is one-to-one. f(CA)=book,f(NH)=table,f(IL)=chair,f(OH)=fork,f(SC)=road.,
b) Define a function g from A to B that is not one-to-one. g(CA)=book,g(NH)=book,g(IL)=book,g(OH)=fork, g(SC)=road.,
c) Define a function h from A to B that is onto. h(CA)=book,h(NH)=table,h(IL)=chair,h(OH)=fork,h(SC)=road,h(WC)=car
d) Define a function Î± from A to B that is not onto. For example Î± (X)=book for all X in A.
e) Define a function Î² from B to A that acts as the inverse of the function f that you created in part a) of this problem.
Answer: Î²(book)=CA; Î²(table)=NH, Î²(chair)=IL, Î²(fork)=OH, Î²(road)=SC.

1. Prove the summation formula for all natural numbers n using indu...