(4) Consider the following cross products

T_a = [a, a+2] x [a, a+2], S_a = [a,2a] x [a, 2a], R_a= [-, ] x [ -, ]

(a) Illustrate one example of each of T_a , S_a and R_a. Also indicate the 4 vertices for each

on the diagram.

(b) (i) Illustrate the two unions T_a, aN and S_a, aN

(ii) Illustrate the three unions T_a, aR⁺ and S_a, aR⁺ and R_a, aR⁺

and describe the results without using the T_a's , S_a's or R_a's

(c) Illustrate R_a, a [1/2 , 2], and also indicate in the illustration, the intersection

R_a, a[1/2, 2]

(d)(i) Illustrate S_a, a [2,3], and also indicate S_a,a[2,3], and describe both

the union and intersection without using the sets S_a

(ii) Prove the description results of (i)

please answer all parts with good reasoning of what is going on. like if you were teaching it.

Comments/hints

(4)(a) 3 nicely drawn boxes with the X and Y axes indicated along with the vertices (corners) labelled. Do it on 3 different illustrations which can be fairly small at this stage.

(b)(i) Now draw larger diagrams which indicates the collection of the S_a's and T_a's

for a's natural numbers

(ii) R⁺ denotes all positive real numbers. Now the boxes move 'continuously' so examine how the 4 corners (for each case T_a's, S_a's and R_a's) move and use that to come up with description of the unions which do not use the boxes.

(c) just cut down the larger version from b(ii) for the R_a's, 1/2. For the intersection, shade it darker within the union. (here < is less than equal to)

(d) (i)Again cut down the b(ii) version for the S_a's, for 2a's.

(ii) again the proofs will involve the method indicated for question (2) start with

x S_a, a[2,3] do steps, with reasons, end up with x (the set you described without using the S_a's). Then do the converse (which amounts to going back

but needing different steps and reasons).

Then repeat the method for the intersection (which is probably easier than for the union).