Discrete Mathematics Answers

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

Define partial order and total order of relations

Let f and g be functions from the positive real numbers to positive real numbers defined by f(x) = [2x]

g(x) = x².2 Calculate f o g and g o f.

Define a bijective function. Explain with reasons whether the following functions are bijective or not. Find also the inverse of each of the functions.

             i.    f(x) = 4x+2, A=set of real numbers

             ii.   f(x) = 3+ 1/x, A=set of non zero real numbers

             iii.  f(x) = (2x+3) mod7, A=N7

Suppose that ab and c are sets such that A ⊆ b and b ⊆ c show that A ⊆ c

write the following sets by listing their elements between braces