Tutorial06-solutions

# Prove that either z or z note this is a problem that

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Unformatted text preview: os cos θ sin θ ρ2 sin 4 0 0 π 2 π 2 cos3 36 0 cos3 θ sin3 θ sin sin6 cos θ sin θ θ · 0 1 3 ) · 0 4 cos3 sin7 = 0 1 2 θ= π 2 3 3 36 (1 − θ= 0 π 2 36 ρ 0 (1 − 2 ) 7 = 36 0 11 − 46 1 1 − 8 10 = 3 40 :) Questions on logical thinking E Consider a rectangle [ smaller rectangles as [×[ ×[ . Assume that it can be partitioned into = ∪ =1 [ ×[ where ( )×( )∩( )×( ) = ∅ for = . Assume also that for each = 1 we either have − ∈ Z or − ∈ Z. Prove that either − ∈ Z or − ∈ Z. Note: This is a problem that looks quite diﬃcult, but there is a very elegant but unexpected solution. This elegant solution is quite clever and not easy to come up with, so in order to help you along, I have outlined some steps you can follow below. (a) Show that B A sin 2π ( + α) (b) Show that − sin 2π ( + β) = 0 for all α ∈ R if and only if B − A is an integer. ∈ Z or − ∈ Z if and only if = 0 for all α β ∈ R. [ ×[ (c) With the small rectangles deﬁned above, what is sin 2π ( + α) · sin 2π ( + β) ×[ [ ? (d) Hence, what is [ ×[ sin 2π ( + α) · sin 2π ( + β) 9 ? sin 2π ( + α) · (e) The result can now be concluded. (...
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