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Essential Concepts

• Sets are collections of distinct elements that can be described or listed within curly brackets. Sets can contain various elements such as numbers, objects, or even other sets.
• A subset is a set that consists of elements from another set, where every element in the subset is also found in the original set. Subsets can have fewer elements than the original set or be equal to it.
• Set operations are used to combine sets in various ways. They include union, intersection, complement, and difference.
• Logic, including Boolean logic, is a fundamental branch of mathematics that deals with reasoning and evaluating the truth or falsehood of statements using logical operators such as “AND”, “OR”, and “NOT”, enabling precise analysis and decision-making.
• Conditional statements are like “if-then” statements. They tell us what will happen based on a certain condition. We use them to figure out what will result from a given situation or rule.
• Truth tables are a way to organize and show all the possible outcomes for different combinations of true and false statements. They help us understand how logical operations like “AND,” “OR,” and “NOT” work by showing us the results for each combination. They are like a chart that tells us what happens when we combine different statements together.
• DeMorgan’s Laws are rules that help us simplify logical expressions. They tell us that when we have a statement that says “NOT” followed by a combination of “AND” or “OR,” we can flip the operation and change the sign of each part.
• A logical argument is a claim where a set of premises support a conclusion. There are two types of arguments: inductive (using specific examples to propose a general conclusion) and deductive (using general statements to propose a specific situation). An inductive argument provides evidence but doesn’t prove the conclusion, while a deductive argument is valid if the conclusion logically follows from true premises and sound if the premises are factually true.
• Venn or Euler diagrams are visual tools used to understand relationships between sets or groups. They help us see how different sets overlap or intersect, making it easier to analyze and evaluate arguments based on the information presented.
• Truth tables are tables that show all possible combinations of true and false statements and their outcomes. By using truth tables, we can understand how logical operations like “AND,” “OR,” and “NOT” work and evaluate the validity of arguments based on these logical rules.
• A logical fallacy is a mistake in reasoning or an error in an argument that makes it unreliable or misleading. The 9 types of logical fallacies are: ad hominem, appeal to ignorance, appeal to authority, appeal to consequence, false dilemma, circular reasoning, straw man, post hoc, and correlation implies causation.

Glossary

Boolean logic

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

cardinality

number of elements in a set

complement of a set

contains everything that is not in the universal set [latex]A[/latex]

difference of two sets

the list of all the elements that are in one set but not present in the other

existential quantifiers

states that a set contains at least one element

implication

a logical conditional sentence stating that the antecedent implies the consequence

intersection of two sets

contains only the elements that are in both sets

logical argument

a claim that a set of premises support a conclusion

logical inference

the process of deriving new knowledge or conclusions based on existing knowledge or premises

proper subset

a subset that is not identical to the original set as it contains fewer elements

set

collection of distinct objects, or elements

subset

set that contains only elements from the set [latex]A[/latex], but may not contain all the elements of [latex]A[/latex]

truth table

a table showing what the resulting truth value of a complex statement is for all the possible truth values for the simple statements

union of two sets

contains all the elements contained in either set (or both sets)

universal set

a set that contains all elements of interest

universal quantifiers

states that an entire set of things share a characteristic