Discrete Mathematics Answers

find the transitive closure of (1 3) (5 5) (1 6) (3 3) (5 6)

For divisibility relation on the set {1,3,6,9,12,5,25,125}, draw Hasse diagram. Then find minimal, maximal, greatest and least elements. Then give the topological sort using the divisibility relation

[P ۷ (P ۸ Q)] → R

If the truth value of (negation p ➡️ q) ➡️ ( p v negation r) is false, then what is the truth value negation p ↔️ r ?

For the following, find if there are any errors in the methods of proof given below. List out these errors and write how you would prove/disprove the statements given below. (a) Statement: If n is an integer and n^2 is divisible by 4, then n is divisible by 4. Proof: Consider the number 144, which is a perfect square divisible by 4 ( since 4 × 36 = 144). Now, considering that √ 144 = 12 so n=12. Since 12 is also divisible by 4 (4 × 3 = 12), the statement holds true. Hence, Proved! (b) Statement: Let p and q be integers and r = pq + p + q, then r is even if and only if p and q are both even. Proof: Since p and q are even we can write them as p = 2k1 and q = 2k2. This means - r = 2k1 · 2k2 + 2k1 + 2k2, r = 2(2 · k1 · k2 + k1 + k2), r = 2(k3) Meaning r is an even number. Therefore, the statement above is true.

If the truth value of ( p ➡️ q) v negation r is false, then what is the truth value negation q ↔️r ?

Show that the power set of S={a,b,c} is a poset under set inclusion