Consider any two finite sets A and B. Prove that |A×B|=|A||B|

Expert Solution

Colby Messinger answered 2 years ago

Let A and B be finite sets. Prove the following: (a) |A∪B|=|A|+|B|−|A∩B| (b) |A × B|...

Let A and B be finite sets. Prove the following: (a) |A∪B|=|A|+|B|−|A∩B| (b) |A × B| = |A||B| (c) |{f : A → B}| = |B||A|

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Prove the following using any method you like: Theorem. If A, B, C are sets, then (A ∪ B) \ C = (A \ C) ∪ (B \C) and A ∪ (B \ C) = (A ∪ B) \ (C \ A)

Prove the following statements! 1. If A and B are sets then (a) |A ∪ B|...

Prove the following statements! 1. If A and B are sets then (a) |A ∪ B| = |A| + |B| − |A ∩ B| and (b) |A × B| = |A||B|. 2. If the function f : A→B is (a) injective then |A| ≤ |B|. (b) surjective then |A| ≥ |B|. 3. For each part below, there is a function f : R→R that is (a) injective and surjective. (b) injective but not surjective. (c) surjective but not injective. (d)...

Unless otherwise noted, all sets in this module are finite. Prove the following statements... 1. If...

Unless otherwise noted, all sets in this module are finite. Prove the following statements... 1. If A and B are sets then (a) |A ∪ B| = |A| + |B| − |A ∩ B| and (b) |A × B| = |A||B|. 2. If the function f : A→B is (a) injective then |A| ≤ |B|. (b) surjective then |A| ≥ |B|. 3. For each part below, there is a function f : R→R that is (a) injective and surjective. (b)...

Prove that a disjoint union of any finite set and any countably infinite set is countably...

Prove that a disjoint union of any finite set and any countably infinite set is countably infinite. Proof: Suppose A is any finite set, B is any countably infinite set, and A and B are disjoint. By definition of disjoint, A ∩ B = ∅ In case A = ∅, then A ∪ B = B, which is countably infinite by hypothesis. Now suppose A ≠ ∅. Then there is a positive integer m so that A has m elements...

C++ Given two finite sets, list all elements in the Cartesian product of these two sets.

Averages and variation Consider two data sets A and B. The sets are identical except the...

Averages and variation Consider two data sets A and B. The sets are identical except the high value of the data set B is three times greater than the high value of data set A. (a) How does the median of the two data sets compare? (b) How do the means of the two data sets compare? (c) How do the standard deviations of the two data sets compare? (d) How do the box- and –whisker plots of the two...

A student suggests that for any two sets A and B: Statement 1: AUB' = (B/A)'...

A student suggests that for any two sets A and B: Statement 1: AUB' = (B/A)' Statment 2: AUB' will never contain an element in set B Is each of these statements correct or incorrect? Use and example or visual representation in your explanation please.

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