Please answer the following questions based on the conversation thus far. Requirements: answer must include explain of the reason Please complete exercise 3.9.6:
Use the following definition of absolute value to prove the given statements: If xx is a real number then the absolute value of xx|x||x| is defined by:|x|={x if x≥0−x if x<0|x|={x if x≥0−x if x<0For any real number xx |x|≥0.|x|≥0. Moreover |x|=0|x|=0 implies x=0.x=0.For any two real numbers xx and yy |x|â‹…|y|=|xy|.|x|â‹…|y|=|xy|.For any two real numbers xx and yy |x+y|≤|x|+|y|. For any real number xx |x|≥0.|x|≥0. Moreover |x|=0|x|=0 implies x=0.x=0. For any two real numbers xx and yy |x|â‹…|y|=|xy|.|x|â‹…|y|=|xy|. For any two real numbers xx and yy |x+y|≤|x|+|y|. Use a truth table to prove that ð‘ð‘ → ð‘žð‘ž and ¬ð‘žð‘ž → ¬ð‘ð‘ are logically equivalent. Here we look at multiple quantifiers.Is the statement ∀ð‘¥ð‘¥ ð‘¦ð‘¦ ∈ R ∃ð‘§ð‘§ ∈ R ð‘ ð‘ . ð‘¡ð‘¡. ð‘§ð‘§ = ð‘¥ð‘¥ true or false? Please prove your response. ð‘¦ð‘¦Write the negation of the statement in part a. – it obviously must have the opposite truth value; do you find this statement easier or more difficult to prove? Is the statement ∀ð‘¥ð‘¥ ð‘¦ð‘¦ ∈ R ∃ð‘§ð‘§ ∈ R ð‘ ð‘ . ð‘¡ð‘¡. ð‘§ð‘§ = ð‘¥ð‘¥ true or false? Please prove your response. ð‘¦ð‘¦ Write the negation of the statement in part a. – it obviously must have the opposite truth value; do you find this statement easier or more difficult to prove? Please complete exercise 3.9.6:
Use the following definition of absolute value to prove the given statements: If xx is a real number then the absolute value of xx|x||x| is defined by:|x|={x if x≥0−x if x<0|x|={x if x≥0−x if x<0For any real number xx |x|≥0.|x|≥0. Moreover |x|=0|x|=0 implies x=0.x=0.For any two real numbers xx and yy |x|â‹…|y|=|xy|.|x|â‹…|y|=|xy|.For any two real numbers xx and yy |x+y|≤|x|+|y|.Use a truth table to prove that ð‘ð‘ → ð‘žð‘ž and ¬ð‘žð‘ž → ¬ð‘ð‘ are logically equivalent.Here we look at multiple quantifiers.Is the statement ∀ð‘¥ð‘¥ ð‘¦ð‘¦ ∈ R ∃ð‘§ð‘§ ∈ R ð‘ ð‘ . ð‘¡ð‘¡. ð‘§ð‘§ = ð‘¥ð‘¥ true or false? Please prove your response. ð‘¦ð‘¦Write the negation of the statement in part a. – it obviously must have the opposite truth value; do you find this statement easier or more difficult to prove? For any real number xx |x|≥0.|x|≥0. Moreover |x|=0|x|=0 implies x=0.x=0.For any two real numbers xx and yy |x|â‹…|y|=|xy|.|x|â‹…|y|=|xy|.For any two real numbers xx and yy |x+y|≤|x|+|y|. Is the statement ∀ð‘¥ð‘¥ ð‘¦ð‘¦ ∈ R ∃ð‘§ð‘§ ∈ R ð‘ ð‘ . ð‘¡ð‘¡. ð‘§ð‘§ = ð‘¥ð‘¥ true or false? Please prove your response. ð‘¦ð‘¦Write the negation of the statement in part a. – it obviously must have the opposite truth value; do you find this statement easier or more difficult to prove?
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