Discrete Mathematics Answers

Using letters L, W, C for the component statements, translate the following compound statements into symbolic notation.

L(x): x is a lawyer

W(x): x is a woman

C(x): x is a chemist

i)There are some women lawyers who are chemists.

ii)No woman is both a lawyer and a chemist.

Determine if 1781 is divisible by 3, 6, 7, 8, and 9

 Solve the recurrence relation of  an+4 + 2an+2 + an=0, n greater than or equal to zero .

What is partial ordering?

Using a Truth table, determine the value of the compound proposition (𝑝 ∨ 𝑞) ∧ (￢𝑝 ∨ 𝑟)) → (𝑞 ∨ 𝑟)

Consider the following relation on set B = {a, b, {a}, {b}, {a, b}}: P = {(a, b), (b, {a, b}), ({a, b}, a), ({b}, a), (a, {a})}. 

Which one of the following alternatives represents the range of P (ran(P))?

1. {a, b, {a}, {a, b}}

2. {a, b, {a}, {b}, {a, b}}

3. {a, b, {b}, {a, b}}

4. {a, b, {a, b}}

Use the factor tree to determine the prime factors of 135

A class of 40 students were each to required three text book of Physics, Mathematics and chemistry however 20 students had physics books, 24 had Mathematics books and 22 had chemistry books. 10 students had Physics and mathematics books ,12 students had physics and chemistry books and 16 had only 2 of the three books. if Every students had at least one of the three books, find, a. how many students had all 3 books. b. how many students only mathematics and chemistry books

Suppose U = {1, 2, 3, a, b, c} is a universal set with the subset A = {a, c, 2, 3}. Let R = { (a, a), (a, c), (3, c), (3, a), (2, 3), (2, a), (2, c), (c, 2) } be a relation on A. Answer questions 11 & 12 by using the given sets A, U and the relation R.

Question 11

Which one of the following statements regarding relation R is true? 1. R is a weak total order. 2. R is a strict total order. 3. R is a weak partial order. 4. R is not an equivalence relation.  Question 12 Which ordered pairs should be removed from relation R in order for the changed relation R1 (say) to be a strict partial order? 1. only (2, c) 2. only (a, a) 3. (a, a) & (c, 2) 4. (a, a) & (2, c) 

Consider the following relation on set B = {a, b, {a}, {b}, {a, b}}: P = {(a, b), (b, {a, b}), ({a, b}, a), ({b}, a), (a, {a})}.

Which one of the following sets is a partition S of B = {a, b, {a}, {b}, {a, b}}? 1. {{a, b, {a}, {b}}, {{a, b}}} 2. {{a}, {b}, {a, b}} 3. {{a, b, {a}}, {{a}, {b}, {a, b}}} 4. {a, b, {a}, {b}, {a, b}} (A partition of the given set B can be defined as a set S = {S1, S2, S3, …}. The members of S are subsets of B (each set Si is called a part of S) such that a. for all i, Si =/ 0/ (that is, each part is nonempty), b. for all i and j, if Si =/ Sj, then Si  Sj = 0/ (that is, different parts have nothing in common), and c. S1  S2  S3  … = B (that is, every element in B is in some part Si). It is possible to form different partitions of B depending on which subsets of B are formed to be elements of S.