How to Chart Truth Tables in High School Math

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Last Updated: July 12, 2023

When you are first introduced to logic in high school, truth values are explained to you through a set of truth tables so that you have a basis for later knowledge to work from. However, while seeing the word truth table might scare you it's the basis of knowledge into what a computer thinks to make their programs become successfully run. But if you find that to be too overwhelming to understand, this article will explain these details to you.

Part 1

Part 1 of 3:

General Charting

1
Define the workings of logic.
- Logic is the study of reasoning and is based on sentence structure.
- Math sentences state facts to complete ideas and so contain facts about subjects and predicates that can be judged to be either True (T) or False (F). Sentences contain statements but have no true variables and therefore can't have both truth values at the same time.
- Truth tables are an expression of symbols showing all possible truth values for a set of sentences.
2
Define several types of logical ideas.
- Negations change the value of a truth value/truth statement. Negations act like a switch, and are formed by adding the word "not" and are shown by the tilde character ~ before the symbol letter.
  - A double negation removes the falsity of the statement and turns these statements' truth values to true.
- Conjunctions combine two sentences together. They represent the and information and are joined together with a ∧ symbol between the two event statements.
  - These truth tables contain more than the two lines since they join two statements alone forming three columns of details - 2 for the values themselves and 1 for the conjunction.
  - In conjunctions, your constructed statements contain both parts of your conclusion as you'll be explained later.
- Disjunctions combine two sentences (with possibilities). They represent the spoken or and are joined together by a ∨ symbol between the two event symbols.
  - The event p ∨ q is true when either part of the sentence is true.
  - However, there's also an exclusive or which is a disjunction that is true only when one of the two is true and not the other, but high school classes may not discuss much (or go into many details) about that case.
- Conditionals form compound sentences that take the form if...than and are joined together with a right-pointing ray (→) between the parts of the sentence.
  - Think of these statements as "steps"; if one thing doesn't happen, then something else will happen instead.
  - Conditionals have two parts: a premise (or hypothesis, antecedent) and a conclusion (or consequent), with the premise written first.
  - A change to the conditional order will change the truth values, as this is later considered a converse.
  - Sometimes, conditionals aren't written with explanations of if..then together, but may be written out in words that replicate this format, and are called a form of hidden conditionals. (E.g. When this assignment gets completed, it should be turned in should become If this assignment gets completed, then it should be turned in.)
- Biconditionals are compound sentences that are formed by a conjunction of a conditional and its converse (or reverse-formed conditional). They are read as "if and only if" and are shown as "↔".
  - Biconditionals are true when both statements are true or both false.
3
Learn special situations. These special situations involve learning inverses, converses, and contrapositives.
- Conditional inverses are formed by negating both the premise and conclusion statements and are read as either "if not p, then not q" or "as not p implies not q."
- Conditional converses are formed by merely switching the hypothesis and conclusion so that it's read "q implies p" or "if q then p".
- Conditional contrapositives are formed by negating both the hypothesis and conclusion as well as switching them so that it's read "not q implies not p" or "if not q then not p."
4
Learn that letters indicate the statement that has truth values. They act much like algebraic symbols holding truthfulness or falsity data. If you find ones in the problem, use those. If not, you will want to find symbols to designate each statement in the compound mathematical sentence.
- Textbooks often mention what they advise they will use early on in the chapter containing logic and truth table creation details - such as p, q, or similar.
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Part 2

Part 2 of 3:

Charting Tables

1
Read the problem you must solve. Determine the statements you must solve. If you don't have each statement expressed as one-letter symbols, express each statement as such. Read your problem and determine what portions you know or what you can easily determine.
2
Write down the problem for the truth table to be solved somewhere on your paper. Write down your statements and symbols as they are expressed on the page.
3
Set up a columnar table. Express each column as its initial truth values. Except a negation truth table, each table must contain two or more columns of four truth values - for four different scenarios, where each row contains one truth value. Row one must contain values which are True, True; Row two must contain values of True, False; Row three must contain False, True; and Row four must contain values of False, False.
- Often books simplify these truth values as T for True and F for False and you must follow the way it is written in the text. (If it is written out in full, continue writing it out in full. But if it's shortened, write it in its shortened way where it needs to go.)
4
Write up truth values for negations, if any exist in the table to be solved. Base this on the values in the first two columns.
5
Work on problems inside the sets of parentheticals ( ). Solve for any conjunctions, disjunctions, conditionals, or biconditionals within these parentheses. Solve each of the four rows all in one column, just to the right of any other columns in the table.
6
Work on problems with truth tables from negations of any parenthesis - if necessary. If the original statement was never negated directly before a full parenthetical statement, don't worry about negating these statements.
7
Work on problems tying together any items to the left of the conjunction, disjunction, conditional or biconditional with the truth values you found to the right. If necessary, find out the truth values by solving the same way - in terms of parentheticals to the left.
8
Tie the right side of the truth table to the left in the last column's result - to form the last column.
- Some truth tables will be rather large, so write somewhat small, but stay consistent in the location of your column separators.
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Part 3

Part 3 of 3:

Using Truth Tables

1
Learn the types of equivalent statements.
- Equivalent statements are shown in a truth table to have the same truth values as another logical (most often: conditionals) statement. These conditionals might have the same values as their inverse, converse, or contrapositives, or their statements are equivalent to a defined type of logical statement such as their parts in conjunctions, disjunctions, or biconditionals...however, this is not always the case. (Two conditionals can be equivalent, but even then, it's not certain that this is the case either.)
- Common scenarios include converses and inverses being equivalent, as well as conditionals and their contrapositives being equivalent.
2
Learn about tautologies.
- Tautologies are basic known truths in logic. Tautologies are known to have all true truth values.
- However, to see this, you have to work out a truth table. If they are tautologies, every truth value in the final column (where you finish off the truth value for the final statement) will have a True or T value in them.
- Most textbooks tell you to write the word "Tautology" underneath the final which is the tautology. However, while you should follow your teacher's advice regarding how they want you to show a truth table column as being a tautology, showing all values as True (as a life skill) will be more than enough to show that something is a tautology.
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Tips

Solving truth tables will seem tedious and arduous because it's something people shy away from. But as you move through other logic topics, you'll see that it makes the advanced info seem more logical because truth tables factor into upcoming needs to know.

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