3. Page 21, 1.3.9 (b) Now prove the general inclusion exclusion formula given by the expression (1.3.13) P(C1uC2u Ck)=p1-p2+p3-+(-1)k+1pk, p1=P(C1) +P(C2)+...+P(CK), p2=1si
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- Does the Identity law and closure law hold for the following: 1. Addition table for Z9 2. Multiplication table for Z9 3. The following table: x 1 2 4 5 7 8 1 1 2 4 5 7 8 2 2 4 8 1 5 7 4 4 8 7 2 1 5 5 5 1 2 7 8 4 7 7 5 1 8 4 2 8 8 7 5 4 2 1If a (b,v,r,k,lambda) design is symmetric with lambda=1, then there exists an integer n such that the design has parameters (n2+n+1, n2+n+1,n+1,n+1,1). Hint: Solve for b and v in terms of k.Consider two points (x0, y0) and (x1, y1). Prove that the first order Langrange polynonial is equivalent to linear interpolation.
- Does closure law hold for the following: 1. Addition table for Z9 2. Multiplication table for Z9 3. The following table: x 1 2 4 5 7 8 1 1 2 4 5 7 8 2 2 4 8 1 5 7 4 4 8 7 2 1 5 5 5 1 2 7 8 4 7 7 5 1 8 4 2 8 8 7 5 4 2 1This question(attached) is related to discrete math and revolves around the Principle of Inclusion/Exclusion. How to apply this principle for the given problem?A. Show that both ℝ and ℝ2 may be covered by countably many open balls
- show that Aut (Z_2 *Z_3) is isomorphic to Aut(Z_2)*Aut (Z_3)Show that R3 is spanned by S = {(1, 1, 1), (1, −1, 0), (0, 1, −1)}.Determine the number of northeast lattice paths from (0, 0) to (4, 4) with unit steps E = (1, 0) and N = (0,1) that pass through neither (1, 1) nor (3, 3). Hint: Inclusion-Exclusion.