MAT 299 Southern New Hampshire University MOD3 De Morgans Law Worksheet Complete the following problems from the attached document. Please note that a PDF

MAT 299 Southern New Hampshire University MOD3 De Morgans Law Worksheet Complete the following problems from the attached document. Please note that a PDF version is provided to ensure that your word processing software has the appropriate math symbols and does not make inappropriate substitutions.Be sure to show your work, including detail with explanations. You can use these templates to help write your proofs and notation in your homework using Equation Editor. MAT 299 Module Three Homework
General:
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Before beginning this homework, be sure to read the textbook sections and the material
in Module Three.
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Type your solutions into this document and be sure to show all steps for arriving at your
solution. Just giving a final number may not receive full credit.
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You may copy and paste mathematical symbols from the statements of the questions
into your solution. This document was created using the Arial Unicode font.
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These problems are proprietary to SNHU COCE, and they may not be posted on any
non-SNHU website.
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The Institutional Release Statement in the course shell gives details about SNHU’s use
of systems that compare student submissions to a database of online, SNHU, and other
universities’ documents.
SNHU MAT299
Page 1 of 3
Module Three Homework
1. Are these statements true or false? The universe of discourse is the set of all people, and
T(x, y) means “x and y are twins.” Explain how you determined your solution.
a. ∀x ¬T(x, x)
b. ∃y T(y, y)
c. ∃x ∀y T(x, y)
d. ∀x ¬∃y T(x, y)
e. ∃x ¬∃y T(x, y)
f.
∀x ¬∀y T(x, y)
This is similar to examples and exercises in Section 2.1 of your SNHU MAT299 textbook.
2. Are these statements true or false? The universe of discourse is ℕ. Explain how you
determined your solution.
a. ∀x(5 < x < 10 → ∃a∃b∃c(a2 + b2 + c2 = x)). b. ∃!x((x – 4)2 = 36). c. ∃!x((x – 11)2 = 49). d. ∃x∃y ( (x ≠ y) ∧ ((x – 4)2 = 36) ∧ ((y – 4)2 = 36) ). This is similar to examples and exercises in Section 2.2 of your SNHU MAT299 textbook. 3. Show that ¬∀x∈A ¬P(x) is equivalent to ∃x∈A P(x). This is similar to t examples and exercises in Section 2.2 of your SNHU MAT299 textbook. SNHU MAT299 Page 2 of 3 Module Three Homework 4. Analyze the logical forms of the following statements. The universe of discourse is ℝ. a. The identity element for multiplication is 1. b. Every positive real number has a positive multiplicative inverse. c. No positive real number has a negative multiplicative inverse. This is similar to examples and exercises in Section 2.2 of your SNHU MAT299 textbook. 5. Let J = {2, 3, 4}, and for each j ∈ J, let Aj = {j, j + 1, j + 2, 2j, 3j}. a. List the elements of all the sets Aj, for j ∈ J. Explain how you determined these sets. b. Find ∩j∈J Aj and ∪j∈J Aj. Explain how you determined these sets. This is similar to examples and exercises in Section 2.3 of your SNHU MAT299 textbook. 6. Show the following for any two sets A and B. a. ℘(A) ∪ ℘(B) ⊆ ℘(A ∪ B). b. Is ℘(A) ∪ ℘(B) = ℘(A ∪ B)? Either provide a proof to show that this is true or provide a counterexample to show that this is false. This is similar to examples and exercises in Section 2.3 of your SNHU MAT299 textbook. SNHU MAT299 Page 3 of 3 Module Three Homework Purchase answer to see full attachment