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Write a predicate calculus statement that involves a universal and an existential quantifier. The domains for each quantifier should be one of the numeric sets, the natural numbers, the integers, the rational numbers or the real numbers. Indicate whether the statement is true or false. If it is false, provide a counterexample. Write the negation of the original statement. If the negation is false, provide a counterexample.

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Propose a rule of inference that includes at least three independent variables, and at least two premises, each containing at least one logical operation. The conclusion must contain at least two logical operations. Using a truth table, assess whether your proposal is a valid rule of inference.

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Provide an example of a proof by mathematical induction. Indicate whether the proof uses weak induction or strong induction. Clearly state the inductive hypothesis. Provide a justification at each step of the proof and highlight which step makes use of the inductive hypothesis.

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Choose a real world example that makes use of a Venn diagram that involves at least three sets. Explain why a Venn diagram is a useful for this example. Discuss the significance of the intersection and union of the various sets.

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Give an example of a partial order relation that is defined on a finite set that contains at least ten elements. State what the set is and how the partial order is defined. Show that the relation satisfies the three properties required of partial order. Draw a directed graph for that partial order.

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Give an example of a function that is defined on the set of integers that is not a one-to-one function. Identify two values that are mapped to the same value to demonstrate that it is not one-to-one. Give an example of a function that is defined on the set of rational numbers that is not an onto function. Identify one value that is not mapped to, to demonstrate that it is not onto. Give an example of a function defined on the real numbers that is a one-to-one correspondence. Define the inverse of that function.

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Give a real-world example of the inclusion/exclusion principle that involves at least two finite sets. Specify values for three of the following four values: the size of the first set, the set of the second set, the size of the union and the size of the intersection. Apply the inclusion/exclusion principle to determine that value of the one value that you did not specify.

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Give an example of a connected undirected graph that contains at least twelve vertices that contains at least two circuits. Draw that graph labeling the vertices with letters of the alphabet. Determine one spanning tree of that graph and draw it. Determine whether the graph has an Euler circuit. If so, specify the circuit by enumerating the vertices involved. Determine whether the graph has an Hamiltonian circuit path. If so, specify the circuit by enumerating the vertices involved.

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