Discrete Mathematics Answers

(a) Let S be a set, and define the set W as follows:
Basis: ∅ ϵ W.Recursive Definition: If x ϵ S and A ϵ W, then {x} U A ϵ W. Provide an explicit description of W, and justify your answer.

Determine the truth value of each of the quantified statements below if the domain consists of R.
(a)∃x(x^5 = -1)
(b)∃x(x^6< x^4)
(c)∀x((-x)4=x^4)
(d)∀x(2x > x)

Consider the scenario of Knights and Knaves. You visited that island and met five

people there. Suppose those were A, B, C, D, E [10]

 A: B is a knight.

 B: E is a knight.

 C: A is a knave.

 D: There are at most two knights among us.

 E: No more than...no no no. [then E Looks at C and] Are we not knights and do not

stand for things that are truth and based on jstice? [then E Turns back to you and

said] Welcome dear guest, A and B are the only two knaves among us.

Tell which is knight and which one is knave (Using truth table or rules).

Transform by making change of variable j = i-4.

Suppose S is a set containing 5 elements, and that ⪯ is a total ordering of S. Draw the Hasse diagram for ⪯ (no need to label the vertices in your diagram).

Definition of Function, Domain, Codomain and Range, Well defined function, Types of functions(Injective ,Surjective, Bijective, Composite)

) We have the following set of information about a computer program, find the mistake in the program using Rules of Inferences.
i. Either a variable is not declared or there is a syntax error in the fifth line.
ii. If there is a syntax error in the fifth line, then there is a missing semicolon or there is a mistake in variable name.
iii. There is not a missing semicolon.
iiii. There is a mistake in variable name.

) We have the following set of information about a computer program, find the mistake in the program using Rules of Inferences.
i. Either a variable is not declared or there is a syntax error in the fifth line.
ii. If there is a syntax error in the fifth line, then there is a missing semicolon or there is a mistake in variable name.
iii. There is not a missing semicolon.
iiii. There is a mistake in variable name.

Use mathimatical Induction to prove that
1²+2²+3²+.......n²= n(n+1)(2n+1) divided by 6