Discrete Mathematics Answers

Give an indirect proof of the theorem; “If 𝑛 is an integer and 

           n³ +13is odd, then n is even.”?  

Check whether the relation R on the set S = {1, 2, 3} is an equivalent relation where:                                           

 𝑅 = {(1,1), (2,2), (3,3), (2,1), (1,2), (2,3), (1,3), (3,1)}. Which of the following properties R has: reflexive, symmetric, anti-symmetric,  transitive? Justify your answer in each case?

Let 𝑆 = {𝑎, 𝑏, 𝑐} and 𝑅 = {(𝑎, 𝑎), (𝑏, 𝑏), (𝑐, 𝑐), (𝑏, 𝑐), (𝑐, 𝑏)}, find [𝑎], [𝑏] and [𝑐] (that is the equivalent class of a, b, and c). Hence or otherwise  find the set of equivalent class of 𝑎, 𝑏 and 𝑐?

Let R be the relation on the set A = {1, 2, 3, 4, 5, 6, 7} defined by  the rule (a b R, ) if the integer product of (ab)        is divisible by 4. List the elements of R and its inverse?  

verify each of the following equivalences using basic equivalences

1)((P∧Q∧R)→S∧(R→(P ∨ Q ∨ S))≡R∧(P↔Q)→S

2)((P∧Q)→R)∧(Q→(S∨R))≡Q∧(S→P)→R

Generate up to the seventh and nineth rows of the Pascal triangle.

Prove by induction that P(n): 2+3+3...+n=n(n+1)/2 Æn ≥ 1

Give a contrapositive proof of the theorem; "If n is an interfer and 3n + 2 is even, then n is even."?

verify each of the following equivalences using basic equivalences

1)((P∧Q∧R)→S∧(R→(P ∨ Q ∨ S))≡R∧(P↔Q)→S

2)((P∧Q)→R)∧(Q→(S∨R))≡Q∧(S→P)→R

Draw the Hasse diagram for divisibility on the set, {1, 2, 3, 4, 5, 6, 7, 8}?