Solved: 2. What does it mean for two propositions to be logically equivalent?

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

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}…

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

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. …

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 …

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…

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: B-C C x A P(A) |P (B)|

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

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} …

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.

If p → q is false, can you determine the truth value of (~p)˅(p ↔ q)? Explain your answer. If p → q is true, can you determine the truth value of ~(p → q)˄~p? Explain your answer.

Prove the equivalence of the following in three different ways (truth table, simplification, each is a logical consequence of the other): p → (q ∨ r) ≡ (p ∧ ~q) → r.

Use the properties to verify the logical equivalences in the following. Supply a reason for each step. a. (p ∧∼ q) ∨ p ≡ p b. p ∧ (∼ q ∨ p) ≡ p c. ∼ (p ∨∼ q) ∨ (∼ p ∧∼ q) ≡ ∼ p d. ∼ ((∼ p ∧ q) ∨ (∼ p ∧∼ q)) ∨ (p ∧ q) ≡ p e. (p ∧…

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} …

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 …

If 𝑝 → 𝑞 is false, can you determine the truth value of (~𝑝)˅(𝑝 ↔ 𝑞)? Explain your answer. If 𝑝 → 𝑞 is true, can you determine the truth value of ~(𝑝 → 𝑞)˄~𝑝? Explain your answer.

a) 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.

Find the graph of the function f(n)=1-n from integers to integer

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 gra…

X=factors of 30 R=x/y draw hasse diagram

A. Write each of the following sets by listing their elements between braces. 2. {3x+2:xϵZ} 4. {xϵN:-2

A. Write out the indicated sets by listing their elements between braces. 2. Suppose A={π,e,0} and B={0,1} (a) AxB (b) BxA (c) AxA (d) BxB (e) ØxB (f) (AxB)xB (g) Ax(BxB) (h) AxBxB 4. {nϵZ:2

A. List all the subsets of the following sets. 2. {1,2,Ø} 8. {{0,1},{0,1,{2}},{0}} B. Write out the following sets by listing their elements between braces. 12. {X:X ⊆{3,2,a} and │X│=1 C. Decide if the following statement…

A. Find the indicated sets. 2.ℙ({1,2,3,4}) 6. ℙ({1,2})x ℙ({3}) 10. {Xϵ ℙ({1,2,3}):│X│≤1} B. Suppose that │A│=m and │B│=n. Find the following cardinalities. 14. │ ℙ(ℙ(A))│ 18. │ ℙ(Ax ℙ(B))│

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 …

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Let A = {1, 2, 3, 4, 5, 6, 7} and R = {(x, y) | x –y is divisible by 3} Show that R is an equivalence relation. Draw the graph of R.

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}…

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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. …

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Let R be the relation on the set A = {1,2,3,4,5,6,7} defined by the rule (a,b) equivalent to R, if the integer (a-b) is divisible by 4, List the elements of R and its inverse

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Q1. a) Let 𝑈 = {𝑥: 𝑥 ∈ 𝑍, 1 ≤ 𝑥 ≤ 12}, 𝐴 = {2𝑥: 𝑥 ∈ 𝑈 𝑎𝑛𝑑 𝑥 𝑖𝑠 𝑎 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒 𝑜𝑓 4}, 𝐵 = {𝑥: 𝑥 ∈ 𝑈, 𝑥 𝑑𝑖𝑣𝑖𝑑𝑒𝑠 2} 𝑎𝑛𝑑 𝐶 = {𝑥: 𝑥 ∈ 𝑈, 𝑥2 ≤ 16}. i) List the elements belong to the sets A, B and C respectively. (3 marks) ii) Find 𝐶 − (𝐴̅ ∩…

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Read and follow instructions.

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2. What does it mean for two propositions to be logically equivalent?

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