Math in Politics: Background You’ll Need 1

  • Engage in logical reasoning and solve basic mathematical problems

Logic

Logic is a systematic way of thinking that allows us to deduce new information from old information and to parse the meanings of sentences. You use logic informally in everyday life and certainly also in doing mathematics. Logic is, basically, the study of valid reasoning.

For example, suppose you are working with a certain circle, call it “Circle [latex]X[/latex],” and you have available the following two pieces of information.

  1. Circle [latex]X[/latex] has radius equal to [latex]3[/latex].
  2. If any circle has radius [latex]r[/latex], then its area is [latex]\pi{r}^{2}[/latex] square units.

You have no trouble putting these two facts together to get:

  1. Circle [latex]X[/latex] has area [latex]9\pi[/latex] square units.

You are using logic to combine existing information to produce new information. Since a major objective in mathematics is to deduce new information, logic must play a fundamental role. This chapter is intended to give you a sufficient mastery of logic.

Boolean Logic

We can often classify items as belonging to sets. If you went the library to search for a book and they asked you to express your search using unions, intersections, and complements of sets, that would feel a little strange. Instead, we typically using words like “and,” “or,” and “not” to connect our keywords together to form a search. These words, which form the basis of Boolean logic, are directly related to set operations with the same terminology.

Boolean logic

Boolean logic combines multiple statements that are either true or false into an expression that is either true or false.

  • In connection to sets, a boolean search is true if the element in question is part of the set being searched.

Suppose M is the set of all mystery books, and C is the set of all comedy books. If we search for “mystery”, we are looking for all the books that are an element of the set M; the search is true for books that are in the set.

When we search for “mystery and comedy” we are looking for a book that is an element of both sets, in the intersection. If we were to search for “mystery or comedy” we are looking for a book that is a mystery, a comedy, or both, which is the union of the sets. If we searched for “not comedy” we are looking for any book in the library that is not a comedy, the complement of the set C.

Connection to set operations

[latex]\begin{array}{r@{\hfill}l}
A \text{ and } B && \text{elements in the intersection } A \cap B \\
A \text{ or } B && \text{elements in the union } A \cup B \\
\text{ Not } A && \text{elements in the complement } A^c \\
\end{array}[/latex]
Notice here that or is not exclusive. This is a difference between the Boolean logic use of the word and common everyday use. When your significant other asks “do you want to go to the park or the movies?” they usually are proposing an exclusive choice – one option or the other, but not both. In Boolean logic, the or is not exclusive – more like being asked at a restaurant “would you like fries or a drink with that?” Answering “both, please” is an acceptable answer.
Suppose we are searching a library database for Mexican universities. Express a reasonable search using Boolean logic.

In most internet search engines, it is not necessary to include the word and; the search engine assumes that if you provide two keywords you are looking for both. In Google’s search, the keyword or has be capitalized as OR, and a negative sign in front of a word is used to indicate not. Quotes around a phrase indicate that the entire phrase should be looked for. The search from the previous example on Google could be written:

Mexico university -“New Mexico”

Conditional Statements

Beyond searching, Boolean logic is commonly used in spreadsheet applications like Excel to do conditional calculations. A statement is something that is either true or false. A statement like [latex]3 < 5[/latex] is true; a statement like “a rat is a fish” is false. A statement like “[latex]x < 5[/latex]” is true for some values of [latex]x[/latex] and false for others. When an action is taken or not depending on the value of a statement, it forms a conditional.

statements and conditionals

A statement is either true or false.

A conditional is a compound statement of the form: “if [latex]p[/latex] then [latex]q[/latex]” or  “if [latex]p[/latex] then [latex]q[/latex], else [latex]s[/latex].”

In common language, an example of a conditional statement would be “If it is raining, then we’ll go to the mall. Otherwise we’ll go for a hike. ”The statement “If it is raining” is the condition—this may be true or false for any given day. If the condition is true, then we will follow the first course of action, and go to the mall. If the condition is false, then we will use the alternative, and go for a hike.

Types of Logical Fallacies

In the previous Learn-It, we saw that logical arguments can be invalid when the conclusion is not true, when the premises are insufficient to guarantee the conclusion, or when there are invalid chains in logic. There are a number of other ways in which arguments can be invalid, a sampling of which are given here.

The ideas sampled on this page are classic and are often found in irresponsible advertising, politics, and in social media. A good way to understand them is to see as many examples of them in the world as you can. Look at social media arguments that appeal to emotion and sentiment. Political candidates, too, sometimes employ one or more of these to favorably manipulate situations.

types of logical fallacies

  1. Ad hominem: An ad hominem argument attacks the person making the argument, ignoring the argument itself.
  2. Appeal to ignorance: This type of argument assumes something it true because it hasn’t been proven false.
  3. Appeal to authority: These arguments attempt to use the authority of a person to prove a claim. While often authority can provide strength to an argument, problems can occur when the person’s opinion is not shared by other experts, or when the authority is irrelevant to the claim.
  4. Appeal to consequence: An appeal to consequence concludes that a premise is true or false based on whether the consequences are desirable or not.
  5. False dilemma: A false dilemma argument falsely frames an argument as an “either or” choice, without allowing for additional options.
  6. Circular reasoning: Circular reasoning is an argument that relies on the conclusion being true for the premise to be true.
  7. Straw man: A straw man argument involves misrepresenting the argument in a less favorable way to make it easier to attack.
  8. Post hoc (post hoc ergo propter hoc): A post hoc argument claims that because two things happened sequentially, then the first must have caused the second.
  9. Correlation implies causation: Similar to post hoc, but without the requirement of sequence, this fallacy assumes that just because two things are related one must have caused the other. Often there is a third variable not considered.

Strategies for Reading and Understanding Math Problems

Reading and understanding math problems is an essential skill for successful problem-solving. Let’s explore effective strategies to help you navigate and comprehend math problems with ease. By applying these strategies, you will develop the ability to extract key information, identify problem objectives, and confidently solve mathematical problems.

    1. Read Carefully: Begin by reading the problem statement attentively. Understand the context, identify the problem’s objective, and note any important details or constraints. Pay close attention to numerical values, units of measurement, and keywords that indicate mathematical operations.
    2. Identify the Question: Determine what the problem is asking you to find or solve. Look for phrases like “Find,” “Calculate,” or “Determine.” Understanding the question will guide your approach and help you focus on the relevant information.
    3. Highlight Key Information: Identify and highlight the essential details and quantities provided in the problem. This includes numerical values, units, and any other relevant data. Use labels or variables to represent unknowns or quantities that need to be calculated. Highlighting the key information allows you to focus on what is important information and ignore unimportant distracting information.
    4. Visualize the Problem: Create mental or written representations of the problem to aid your understanding. Draw diagrams, charts, or graphs if appropriate. Visualizing the problem can help you see the relationships between quantities and identify potential solution paths.
    5. Break It Down: Break the problem into smaller parts or steps. Analyze each part individually to understand its purpose and how it contributes to the overall solution. Identify the mathematical operations or concepts required for each step.
    6. Use Problem-Solving Strategies: Familiarize yourself with problem-solving strategies, such as working backward, making a table or chart, using logical reasoning, or applying relevant formulas or equations. Choose the strategy that best suits the problem at hand. We will explore problem-solving strategies more later in this section.
    7. Solve Step-by-Step: Once you have a plan in mind, solve the problem step-by-step. Show your work and perform the necessary calculations, ensuring accuracy and attention to detail.
    8. Check Your Answer: After finding a solution, verify if it makes sense in the given context. Re-read the problem, check your calculations, and assess whether the answer aligns with the question’s requirements. If possible, use estimation or alternative methods to confirm the reasonableness of your answer.
    9. Reflect on the Solution: Take a moment to reflect on the problem-solving process. Evaluate the effectiveness of your strategy and identify any insights or learnings that can be applied to similar problems in the future.

By employing these strategies, you can improve your ability to read and understand math problems, identify relevant information, and approach problem-solving with clarity and confidence. Regular practice with a variety of problems will enhance your skills and deepen your mathematical understanding.

Problem-Solving Strategies

In many problems, it is tempting to take the given information, plug it into whatever formulas you have handy, and hope that the result is what you were supposed to find. Chances are, this approach has served you well in other math classes. This approach does not work well with real-life problems. Instead, problem-solving is best approached by first starting at the end: identifying exactly what you are looking for. From there, you then work backward, asking “what information and procedures will I need to find this?”

How To: Problem-Solving Process

  1. Identify the question you’re trying to answer.
  2. Work backward, identifying the information you will need and the relationships you will use to answer that question.
  3. Continue working backward, creating a solution pathway.
  4. If you are missing any necessary information, look it up or estimate it. If you have unnecessary information, ignore it.
  5. Solve the problem, following your solution pathway.

Now that we have a process for problem-solving, let’s talk about the different approaches we can take to solve a problem.

    • Break It Down: Complex problems can often be daunting. But here’s a secret – they’re just a bunch of simple problems grouped together. Our first approach should always be to break down complex problems into smaller, more manageable parts.
      • For example, if you are asked “”A farmer has chickens and cows in his farm. He counts [latex]50[/latex] heads and [latex]140[/latex] legs. How many chickens and cows does he have?” you may think this sounds super complex but if we break it down we see it isn’t so bad
          • All animals have [latex]1[/latex] head. So, the [latex]50[/latex] heads mean we have [latex]50[/latex] animals.
          • Chickens have [latex]2[/latex] legs, cows have [latex]4[/latex]. So if all [latex]50[/latex] animals were chickens, we would have [latex]100[/latex] legs.
          • But we have [latex]140[/latex] legs, which is [latex]40[/latex] more than [latex]100[/latex]. Since each cow has [latex]2[/latex] extra legs compared to a chicken, the [latex]40[/latex] extra legs mean we have [latex]20[/latex] cows ( [latex]40 ÷ 2 = 20[/latex]).
          • Finally, since we have [latex]50[/latex] animals in total, the remaining [latex]30[/latex] must be chickens.
    • Trial and Error: Sometimes, problems don’t have a clear path to the solution. In these cases, good old trial and error can come to our rescue.
      • For example, if you are asked “What is the value of [latex]x[/latex] in the equation [latex]2^x = 32?[/latex]” it may be tempting to jump right into logarithms, but if you try a few values for [latex]x[/latex] first it may help to find the answer.
          • If [latex]x = 4[/latex], then [latex]2^x = 2^4 = 16[/latex]. Too small.
          • If [latex]x = 5[/latex], then [latex]2^x = 2^5 = 32[/latex]. Bingo!
    • Pattern Recognition: In mathematics, patterns are everywhere! Recognizing these patterns can make problem-solving a breeze.
      • For example, if you are asked “What is the [latex]5[/latex]th term in the sequence: [latex]2, 4, 8, 16,[/latex]…?” looking for a pattern will help solve the problem.
          • Here, we can see that each term is twice the previous one. So, the [latex]5[/latex]th term is [latex]16*2 = 32[/latex].
    • Logical Reasoning: using logical reasoning can be a powerful problem-solving strategy. This involves creating a logical sequence of steps to solve the problem.
      • For example, if you are asked “If all squares are rectangles, and all rectangles have four sides, do all squares have four sides?” you can use logic to find a solution
          • We can logically reason that since every square is a rectangle, and every rectangle has four sides, it must be that every square has four sides too.

Let’s try using our problem-solving process and the approaches we just learned to solve a few examples.

In the first example, we will need to think about time scales, we are asked to find how many times a heart beats in a year, but usually we measure heart rate in beats per minute.

How many times does your heart beat in a year?

The technique that helped us solve the last problem was to get the number of heartbeats in a minute translated into the number of heartbeats in a year. Converting units from one to another, like minutes to years is a common tool for solving problems.

In the next example, we show how to infer the thickness of something too small to measure with every-day tools. Before precision instruments were widely available, scientists and engineers had to get creative with ways to measure either very small or very large things. Imagine how early astronomers inferred the distance to stars, or the circumference of the earth.

How thick is a single sheet of paper? How much does it weigh?

We can infer a measurement by using scaling.  If [latex]500[/latex] sheets of paper is two inches thick, then we could use proportional reasoning to infer the thickness of one sheet of paper.

Try using the problem-solving process and the approaches you learned to solve some questions on your own.