Discrete Mathematics Answers

IfD is a strong digraph of order n such that degv ≥ n for every vertex v of D, then D is Hamiltonian. Show that if the digraph D is not required to be strong, then D need not be Hamiltonian

let A, B and C is a set where A u B = A u C and A n B = A n C. show that B = C

Consider the following statement:

M = Syahmi will get grade A for his mathematic subject if and only if he study the subject every day.

(i) Write each statement in terms of p and q. Then re-write the statement into logical notation.

(ii) State the negation, converse and contrapositive of the above statement, M, in symbolic form.

(iii) Using the answer in (i), determine the truth table of the propositions in statement M.

(iv) Is the statement M a tautology, contradiction or neither? Justify your answer.

(v) Construct a truth table for r ˅ q --> (ṝ ᴧ p) and p ᴧ q ↔ r. Determine whether both proposition is logically equivalent. Justify your answer.

Let relation defined as below:

(i) Write R1 relation as a set of ordered pairs. Draw the arrow diagram and determine whether R1 is a function or not. Explain your answer.
(ii) Hence, determine whether the R1 relation is an equivalence relation or a partial order (or neither).

(iii) Describe how the digraph of the R1 relation be used to determine whether R1 is an equivalence relation. Your answer should include the digraph and detailed description.

(iv) Determine the matrix of the R1 relation (relative to the given orderings). Now, reorder R1 as 3,2,1,4, and determine the new matrix obtained.

(v) Another technique to test for reflexive, symmetric and transitivity is by using the
matrix of relation. Analyze matrix of the R1 relation to determine whether it is an
equivalence relation.

Question 1(a)
Twenty-five different beauty products make the following claim: 13 claims for whitening the skin, 20 claims to soften the skin, 11 claims to reduce wrinkles, 8 claims to both whitening and softening the skin, 5 claims to both whitening and reducing wrinkles, and 6 claims to both softening the skin and reducing the wrinkles.

(i) Give a suitable name for the sets. Indicate the cardinality of each set. Determine and propose set operations that can be performed on the sample data above. Justify your reasons for proposing the set operations.

(ii) Create Venn diagram to represent the set operations that you derive from Question (i) above. Analyze the Venn diagram and state the conclusion that you can derive from the Venn diagram.

(iii) Analyze how many products make all three claims. Justify your answer.





(iv) Analyze how many products claim for whitening the skin but do not claim to reduce wrinkles. Justify your answer.

(v) Give another example of how sets can be applie

Twenty-five different beauty products make the following claim: 13 claims for whitening the skin, 20 claims to soften the skin, 11 claims to reduce wrinkles, 8 claims to both whitening and softening the skin, 5 claims to both whitening and reducing wrinkles, and 6 claims to both softening the skin and reducing the wrinkles.

(i) Give a suitable name for the sets. Indicate the cardinality of each set. Determine and propose set operations that can be performed on the sample data above. Justify your reasons for proposing the set operations.

(ii) Create Venn diagram to represent the set operations that you derive from Question (i) above. Analyze the Venn diagram and state the conclusion that you can derive from the Venn diagram.

(iii) Analyze how many products make all three claims. Justify your answer.





(iv) Analyze how many products claim for whitening the skin but do not claim to reduce wrinkles. Justify your answer.

(v) Give another example of how sets can be applied to group obj

Prove (combinatorially) that:
2*3^0+ 2*3^1*2*3^2 + ... + 2*3^(n-1) = 3^n - 1

Prove:
1. Show that f: R implies R given by f(x)=x^3 is bijective while g:R implies R given by g(x)=((x)^2)-1
2. Let A, B non empty sets f: A implies B. Show that f-1(f inverse) o f is an equivalent relation on A.
3. If f: A implies b is one to one and onto then f-1(f inverse) is also one to one correspondence.
4. Let f: A implies B and g: B implies C be one to one correspondence. Then g o f : A implies C is one to one correspondence.
5. Let f: A implies B and g: B implies C be one to one correspondence. Then (g o f)^-1 (x)=(f^-1 o g^-1) (x) = f^-1(g^-1 (x))

let G and H be graphs. prove that if G and H are sets, then G (inverese) and G circle H are sets.