Use the element method for proving that a set equals the empty set to prove the following statement. (Assume that all sets are subsets of a universal set U.) If U denotes a universal set, then UC = ø. Proof by contradiction: Consider the sentences in the following scrambled list. But, by definition of a universal set, U contains all elements under discussion, and so x E U. So, by definition of complement x E U. Let U be a universal set and suppose Uº = Ø. Let U be a universal set and suppose UC ± Ø. Then there exists an element x in UC. Thus x EU and x € U, which is a contradiction. So, by definition of complement x ¢ U. But, by definition of a universal set, UC contains no elements. We construct the proof by selecting appropriate sentences from the list and putting them in the correct order. 1. Let U be a universal set and suppose UC = Ø. 2. Let U be a universal set and suppose UC = Ø. 3. ---Select--- 4. ---Select--- 5. Thus x e U and x ¢ U, which is a contradiction. 6. Hence the supposition is false, and so UC = ø.

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A:

Q: 14. For all sets A, B, and C, if AC B then A U C CBUC.

A: The detailed explanation is given as below:

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Q: (1) Prove, or come up with as simple a counterexample as you can: For sets A, B, in the context of a…

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Q: n so is {A} but do NOT use an the argu

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Q: For any sets A, B, C  use element argument to prove that  (A-B) - C is a subset of A - C

A: By the element argument we prove the given problem as follows :

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Q: Indirect Proof

A:

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A:

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Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.