Monthly Archives: December 2015

Jail Pass

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In 1996, a 19-year-old man was arrested, and charged with murder. Although, the man claimed he was innocent, he was convicted of the crime, and sentenced to 25 years in prison. In 2006, after serving ten years of his sentence, a man came forward with an incredible story; that he, in fact, was the person that committed the crime, and the wrong man had been in prison for the past ten years.

After an intense investigation, it was concluded that the man was, in fact, telling the truth; he committed the crime. He had too many details of the crime that were never released to the public, and was able to answer questions, that, until now, remained unanswered.

After a short trial, this man was convicted of the crime, and was sentenced to 25 years in prison. The 19-year-old man (now 29), that was wrongly imprisoned, was released the next day. He was issued a formal apology, and was given a check for a measly $300, to help him start a new life, and find a job.

Shortly after the man’s release from prison, a third man came to the police department, and confessed that he, and the now 29-year-old man, committed a robbery, in 1993. So, after being released for only 6 days, the 29-year-old man found himself arrested once again, and back in the court room. After a trial that lasted 5 weeks, and all the evidence was presented, along with a confession from both men, they were both convicted of the robbery.

The third man was sentenced to 17 years in prison for his part, and the 29-year-old man was sentenced to 10 years in prison, for his part in the crime. Now, the 29-year-old man obtains another lawyer, and asks to have his case dismissed. His lawyer presents the argument that he should be released from prison. His sentence is ten years for the robbery; however, he has already spent ten years in prison for a crime that he was acquitted of committing.

The victims of the robbery seek justice, and want both men placed in prison.

What does the judge do? How does he decide the fate of the 29-year-old man? Should the man remain in prison, and serve his ten-year sentence, or should the previous ten years that he served for a crime that he didn’t commit, be applied to his current sentence? It was a sentence for a different crime, but he was wrongly convicted because of a failure in the justice system.

Should the man be required to give away another ten years of his life, or should the system give him a break, and set him free, with time paid?

  • Should the judge give him a jail pass?
  • Would it even be legal, for the judge to give him a jail pass?

Working with Sets

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A set is a well-defined collection of distinct objects. The objects of a set are called elements. A set is well-defined, if it can be easily understood, whether or not, a given object, is an element of that set. Some sets do not have any elements. These sets are called empty sets, or null sets. They are denoted by using the symbol Ø.

N = Ø

A set of digits could be defined as:

D = {1, 2, 3, 4}

The set is labeled D and includes the digits 1 through 4. The left and right braces are used to enclose all of the elements of the set. This makes it easy to understand what is, and is not, an element of the set D. This method of defining a set is called the roster method.

Set-builder Notation is another method that can be used to define a set. An example of the same set, using set-builder notation would be:

D = {x|x is a digit between 1, and 4, inclusive}

This set would be read as “D is the set of all x, such that, x is a digit between 1 and 4, inclusive.” The horizontal pipe | is pronounced “such that.” Therefore, when x|x, is used, it should be read as “x, such that, x.”

When the word “inclusive,” is used, it means that both 1 and 4 are also included in the set; otherwise, a digit between 1 and 4 would be either 2 or 3.

Z = {x|x is a digit between 1, and 4}

In this case, set D (above) would have 4 elements, and set Z, would have 2 elements.

Elements Are Unique

Elements in a set should be unique, i.e., they should not appear more than once in the set. You would not create a set such as: {3, 9, 1, 3}, because the element “3”, appears more than once in the set. The order of the set does not matter. Both {3, 6, 9} and {6, 9, 3} represent the same set.

Equal Sets

When two sets are the same, such as: X = {5, 4, 3}, and Y = {3, 4, 5}, then it can also be said that X = Y. The set of both X and Y are the same, even though the elements are not displayed in the same order in both sets.

Subsets

If one set contains all of the elements of another set, it is called a subset. As an example: P = {4, 9, 0}, and Q = {0, 9, 6, 4, 3}. In this case, P, is a subset of Q, because every element in P, is also an element in Q. The set of Q also contains 4, 9, and 0. When a set is a subset of another set, it is written as: P ⊆ Q.

Intersection

The intersection of a set is defined as: A ∩ B. If a problem uses this format, it’s simply asking for a list of all elements that appear in both set A and set B. If the element “5”, appears in both sets, and all other elements are unique, then the answer would be {5}.

A = {4, 5, 9}; B = {5, 1, 2}; Then A ∩ B = {5}.

Union

The union of a set is defined as: A ∪ B. If a problem uses this format, it’s simply asking for a list of all elements, that appear in either A or B, or both. If an element appears in both sets, it is only listed once in the union. Look at element “8” in the sample:

A = {9, 8, 4}; B = {2, 1, 8}; Then A ∪ B = {9, 8, 4, 2, 1}.

Universal Set

A universal set, written as ∪, is a set that consists of all elements that need to be considered.

Set Complement

The complement of a set, written as Ā (with a line above the set name), is simply a list of all of the elements in the ∪ (universal) set, that are not in set A. In this case, if ∪ = {1, 2, 3, 4, 5, 6, 7, 8, 9}, and A = {6, 7, 9}, then Ā = {1, 2, 3, 4, 5, 8}.