The following are functions. Find the range of each. (Tip: Represent their inputs and outputs as...

  
 
 

  
 

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The following are functions. Find the range of each. (Tip: Represent their inputs and outputs as a table) 1. g: {0, 1, 2, 3, 4, 5} {0, 1, 2, 3, 4, 5} with g(x) = (2x + 1) % 6 2. f {0, 1,2,3,4,5,6} {0, 1, 2, 3, 4, 5, 6} given by f(x) = (3x + 1) % 7 For each of the following functions, determine if the function is injective, whether it is surjective, or whether it is bijective. If any property fails to hold, give a brief explanation why. If it does hold, you do not need to explain why. (Tip: Represent their inputs and outputs as a table) Also, note that these are the same functions as in 04 above. 1. g: {0, 1, 2, 3, 4, 5} → {0, 1, 2, 3, 4, 5} with g(x) = (2x+1) % 6 2. f {0, 1, 2, 3, 4, 5, 6} {0, 1, 2, 3, 4, 5, 6) given by f(x) = (3x+1) % 7 Directions for each of the questions below: If the original statement is in symbols, your answers should be in symbols; if it is in words, the answer should be in clear English as well. For symbolic statements, don't just put in front of the original to form the negation-use De Morgan's laws to simplify. Similarly, for English statements, do no just write "It is not that case that"... to form the negation. 1. For each of the conditional statements below, write the converse, inverse, contrapositive, and negation (fully simplified and clear). (a) If I have coffee, then I am happy. (b) p→q 2. Use De Morgan's laws to state the negations of each of the following (fully simplified and clear): (a) p^ (q vr) (b) It was sunny last week and either school was canceled or we were wearing hats. 1. Determine whether the statement (p V q) → p is a tautology. Show your work and clearly indicate your answer. 2. Determine whether the following statements are logically equivalent or not. Show your work to clearly indicate your answer. PAq and (p→q) The following are functions. Find the range of each. (Tip: Represent their inputs and outputs as a table) 1. g: {0, 1, 2, 3, 4, 5} {0, 1, 2, 3, 4, 5} with g(x) = (2x + 1) % 6 2. f {0, 1,2,3,4,5,6} {0, 1, 2, 3, 4, 5, 6} given by f(x) = (3x + 1) % 7 For each of the following functions, determine if the function is injective, whether it is surjective, or whether it is bijective. If any property fails to hold, give a brief explanation why. If it does hold, you do not need to explain why. (Tip: Represent their inputs and outputs as a table) Also, note that these are the same functions as in 04 above. 1. g: {0, 1, 2, 3, 4, 5} → {0, 1, 2, 3, 4, 5} with g(x) = (2x+1) % 6 2. f {0, 1, 2, 3, 4, 5, 6} {0, 1, 2, 3, 4, 5, 6) given by f(x) = (3x+1) % 7 Directions for each of the questions below: If the original statement is in symbols, your answers should be in symbols; if it is in words, the answer should be in clear English as well. For symbolic statements, don't just put in front of the original to form the negation-use De Morgan's laws to simplify. Similarly, for English statements, do no just write "It is not that case that"... to form the negation. 1. For each of the conditional statements below, write the converse, inverse, contrapositive, and negation (fully simplified and clear). (a) If I have coffee, then I am happy. (b) p→q 2. Use De Morgan's laws to state the negations of each of the following (fully simplified and clear): (a) p^ (q vr) (b) It was sunny last week and either school was canceled or we were wearing hats. 1. Determine whether the statement (p V q) → p is a tautology. Show your work and clearly indicate your answer. 2. Determine whether the following statements are logically equivalent or not. Show your work to clearly indicate your answer. PAq and (p→q)

Answer rating: 100% (QA)

Solutions Step 1 1 The images of the function are g0161g1363g2565g3... View the full answer