Discrete Mathematics Answers

Find simpler statement forms that are logically equivalent to p ⊕ p and (p ⊕ p) ⊕ p.

b) Is (p ⊕ q) ⊕ r ≡ p ⊕ (q ⊕ r)? Justify your answer.

c) Is (p ⊕ q) ∧ r ≡ (p ∧ r) ⊕ (q ∧ r)? Justify your answer.

A. List the members of the following sets

1. {x| x is real numbers and x2 = 1}

2. {x| x is an integer and -4 < x ≤ 3}

B. Use set builder notation to give description of each of these sets.

1. {a, e,i ,o, u}

2. {=2, -1, 0, 1, 2}

C. Let A= (a, b, c), B = (x, y) and C = (0, 1)

Find:

1. A U C

2. C x B

3. B – A

4. (A ∩ C) U B

D. Find these terms of the sequence (An}, where An = 2(3)n + 5

1. A0

2. A5

3. A3

4. 8th term

5. 2nd term

6. Sum of the sequence 

E. Given the following set:

2. X = {-1, 0, 1, 2, 3, 4, 5} defined by the rule (x, y) ∈R if x ≤ y

F. List the elements of R

G. Find the domain of R

H. Find the range of R

I. Draw the digraph

J. Properties of the Relation 

RELATION.

Given the following set:

1. X = {1, 2, 3, 4, 5} defined by the rule (x, y) ∈ R if x + y ≤ 6 

a. List the elements of R

b. Find the domain of R

c. Find the range of R

d. Draw the digraph

e. Properties of the Relation 

Universal set (U) = {-2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, a, b, c, d}

A = {a, b, c}

B = {-2, -1, 0, 1, 2}

C = {2, 4, 6}

find:

1. B-C
2. C x A
3. P(A)
4. |P (B)|

What is the complete proof for INCLUSION-EXCLUSION FORMULA and BOOLE'S INEQUALITY? Step by Step Solutions. The BOOLE'S INEQUALITY that I meant was both:

𝑃(𝐴1 ∪ 𝐴2 ∪ … ∪ 𝐴𝑛 ) ≤ 𝑃(𝐴1 ) + 𝑃(𝐴2 ) + ⋯ + 𝑃(𝐴𝑛) or 𝑃(⋃ 𝐴𝑖 𝑛 𝑖=1 ) ≤ ∑ 𝑃(𝐴𝑖) 𝑛 𝑖=1 .

𝑃(𝐴1 ∪ 𝐴2 ∪ … ∪ 𝐴𝑛 ) ≥ 𝑃(𝐴1 ) + 𝑃(𝐴2 ) + ⋯ + 𝑃(𝐴𝑛) or 𝑃(⋃ 𝐴𝑖 𝑛 𝑖=1 ) ≥ ∑ 𝑃(𝐴𝑖) 𝑛 𝑖=1 .

1. Given the following:

• g: "You can graduate."
• m: "You owe money to the college."
• r: "You have completed the requirements of your major."
• b: "You have an overdue book."

Translate "You can graduate only if you have completed the requirements of your major, you do not owe money to the college, and you do not have an overdue book." into a propositional logic.

2. Show that  are logically equivalent. (15 points)

3. Show the truth table of (truth matrix)  . 

Self - Assessment

A. List the members of the following sets

1. {x| x is real numbers and x2 = 1}

2. {x| x is an integer and -4 < x ≤ 3}

B. Use set builder notation to give description of each of these sets.

1. {a, e,i ,o, u}

2. {=2, -1, 0, 1, 2}

C. Let A= (a, b, c), B = (x, y) and C = (0, 1)

Find:

1. A U C

2. C x B

3. B – A

4. (A ∩ C) U B

D. Find these terms of the sequence (An}, where An = 2(3)n + 5

1. A0

2. A5

3. A3

4. 8th term

5. 2nd term

6. Sum of the sequence

E. Given the following set:

2. X = {-1, 0, 1, 2, 3, 4, 5} defined by the rule (x, y) ∈R if x ≤ y

F. List the elements of R

G. Find the domain of R

H. Find the range of R

I. Draw the digraph

J. Properties of the Relation 

C. RELATION.

Given the following set:

1. X = {1, 2, 3, 4, 5} defined by the rule (x, y) ∈ R if x + y ≤ 6

a. List the elements of R

b. Find the domain of R

c. Find the range of R

d. Draw the digraph

e. Properties of the Relation 

I. SET.

A. List the members of these sets

1. {x | x is a positive integer less than 10}

2. {x | x is an integer such that x2 = 2}

B. Consider the following sets.

Universal set (U) = {-2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, a, b, c, d}

A = {a, b, c}

B = {-2, -1, 0, 1, 2}

C = {2, 4, 6}

Find for the following:

a. A U B

b. 𝐴 ∩ 𝐵

c. B – C

d. C x A

e. 𝐵 ∩

f. 𝐴̅ − 𝐵 

g. C – (𝐵̅ − 𝐴) 

h.  

i. P (A)

j. |𝑃(𝐵)|

) (P ∧ Q ∧ R) ∨ (¬P ∧ R ∧ Q) ∨ (¬P ∧ ¬Q ∧ ¬R