Logic Basics: Fresh Take

  • Combine sets using Boolean logic and proper notation
  • Create and interpret expressions using statements and conditionals
  • Construct and analyze truth tables for complex statements or conditionals
  • Determine the logical equivalence between two statements

Boolean Logic

The Main Idea 

Boolean logic, named after mathematician George Boole, is a system of logic that deals with binary variables (variables that can only take on one of two values, typically denoted as true or false). In the context of set theory, Boolean logic can be used to combine and manipulate sets.

The primary operations in Boolean logic are “and” (corresponding to intersection in set theory), “or” (corresponding to union), and “not” (corresponding to complement). Here’s how they work in the context of sets:

  • And (Intersection): The intersection of two sets [latex]A[/latex] and [latex]B[/latex], represented as [latex]A \cap B[/latex], is the set of elements that are in both [latex]A[/latex] and [latex]B[/latex]. If we consider sets as conditions to be satisfied, the intersection represents the condition where both conditions [latex]A[/latex] and [latex]B[/latex] are true.
  • Or (Union): The union of two sets [latex]A[/latex] and [latex]B[/latex], represented as [latex]A \cup B[/latex], is the set of elements that are in [latex]A[/latex], in [latex]B[/latex], or in both. In terms of conditions, the union represents the condition where either condition [latex]A[/latex] or condition [latex]B[/latex] or both are true.
  • Not (Complement): The complement of a set [latex]A[/latex], represented as [latex]\bar{A}[/latex] or [latex]A'[/latex], is the set of all elements that are not in [latex]A[/latex]. In terms of conditions, the complement represents the negation of condition A.
Describe the numbers that meet the condition: even and less than [latex]10[/latex] and greater than [latex]0[/latex].

You can view the transcript for “Boolean Operators: Pirates vs. Ninjas” here (opens in new window).

Quantified Statements

The Main Idea 

In mathematics, a quantifier is a symbol or word used to express the extent to which a predicate (or proposition) is true over a range of values. There are two main types of quantifiers in mathematics: universal quantifiers and existential quantifiers.

  1. The universal quantifier is typically symbolized by [latex]\forall[/latex] (an inverted letter ‘A’), which stands for “for all” or “for every.” When we use the universal quantifier, we are saying that a certain property or condition holds for all members of a specific set.
  2. The existential quantifier is symbolized by [latex]\exists[/latex] (a backwards letter ‘E’), which stands for “there exists.” When we use the existential quantifier, we are saying that there is at least one member of a set that satisfies a certain property or condition.
Suppose your friend says “One of these six cartons of milk is leaking.” What is the minimum amount of evidence you would need to prove your friend wrong?

You can view the transcript for “Universal and Existential Quantifiers, ∀ “For All” and ∃ “There Exists”” here (opens in new window).

Negating Quantified Statements

In mathematical logic, the process of negating a quantified statement involves changing the quantifier and negating the predicate. The negation of a universally quantified statement results in an existentially quantified statement and vice versa. The exact rules for negating quantified statements are as follows:

  1. Negation of Universal Quantification: If we have a universally quantified statement such as [latex]\forall x, P(x)[/latex] (read as: for all [latex]x[/latex], [latex]P(x)[/latex] is true), the negation of this statement is [latex]\exists x, \neg P(x)[/latex] (read as: there exists an [latex]x[/latex] such that [latex]P(x)[/latex] is not true). Essentially, we are saying that there is at least one case where the proposition does not hold.
  2. Negation of Existential Quantification: If we have an existentially quantified statement such as [latex]\exists x, P(x)[/latex] (read as: there exists an [latex]x[/latex] such that [latex]P(x)[/latex] is true), the negation of this statement is [latex]\forall x, \neg P(x)[/latex] (read as: for all [latex]x[/latex], [latex]P(x)[/latex] is not true). In other words, we are saying that there is no case where the proposition holds.
Write the negation of this statement: “There are no prime numbers that are even.”

Conditional Statements

The Main Idea 

  • A statement is a declarative sentence that is either true or false but not both. For instance, the statement “[latex]p[/latex]: The sky is blue” is a statement that can be true or false.
  • A conditional statement is a compound statement of the form “if [latex]p[/latex] then [latex]q[/latex]” (denoted as [latex]p → q[/latex]), where [latex]p[/latex] and [latex]q[/latex] are statements.

You can view the transcript for “Conditional Statements & Converse Statements | Mathematical Reasoning | Don’t Memorise” here (opens in new window).

Conditional Statements and Excel

Excel provides powerful tools for implementing conditional logic, enabling you to manipulate and analyze data based on specific conditions.

IF Statements: The IF function is a fundamental tool for creating conditional statements in Excel. It tests a condition and returns one result if the condition is true and another if it’s false. The basic syntax is IF(logical_test, value_if_true, value_if_false). For example, =IF([latex]A1>5[/latex],”Greater”,”Smaller”) would return “Greater” if the value in cell [latex]A1[/latex] is greater than [latex]5[/latex], and “Smaller” otherwise.

AND, OR, NOT Functions: You can use the AND, OR, and NOT functions to create more complex conditions. The AND function returns TRUE if all its arguments are true, the OR function returns TRUE if any of its arguments are true, and the NOT function reverses the value of its argument.

Write an Excel command that will create the condition “[latex]A1 < 3000[/latex] and [latex]A1 > 100[/latex]”.

An accountant needs to withhold [latex]15\%[/latex] of income for taxes if the income is below [latex]$30,000[/latex], and [latex]20\%[/latex] of income if the income is [latex]$30,000[/latex] or more. Write an Excel expression that would calculate the amount to withhold.

Truth Tables

The Main Idea 

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

The symbol [latex]\wedge[/latex] is used for and: [latex]A[/latex] and [latex]B[/latex] is notated [latex]A\wedge{B}[/latex] .

The symbol [latex]\vee[/latex] is used for or: [latex]A[/latex] or [latex]B[/latex] is notated [latex]A\vee{B}[/latex] .

The symbol [latex]\sim[/latex] is used for not: not [latex]A[/latex] is notated [latex]\sim{A}[/latex] .

To create a truth table, you need to:

  1. Identify the Propositions: Start by identifying the individual propositions or statements that make up your complex statement.
  2. Create the Table: Draw a table with one column for each proposition and one for the complex statement. The table should have as many rows as there are possible combinations of truth values for the propositions – this is typically [latex]2^n[/latex], where [latex]n[/latex] is the number of propositions.
  3. Fill in the Truth Values: In the proposition columns, fill in all possible combinations of truth and falsehood. Start with alternating truth and falsehood in the first column, then alternate pairs of truths and falsehoods in the second, and so on.
  4. Evaluate the Complex Statement: For each row, evaluate the truth or falsehood of the complex statement based on the truth values of the propositions in that row.

For example, consider the complex statement [latex]P \text{ AND } Q[/latex] (written as [latex]P ∧ Q[/latex] in logic). The truth table would look like this:

[latex]P[/latex] [latex]Q[/latex] [latex]P ∧ Q[/latex]
T T T
T F F
F T F
F F F
Translate “We have carrots or we will not make soup” into symbols. Let [latex]C[/latex] represent “we have carrots” and let [latex]S[/latex] represent “we will make soup.”

Create a truth table for this statement: [latex]\sim A \wedge B[/latex] .

Create a truth table for this statement: [latex](\sim A \wedge B) \vee \sim B[/latex] .

You can view the transcript for “Intro to Truth Tables | Negotiation, Conjunction, and Disjunction” here (opens in new window).

Implications

In logic, an implication, also known as a conditional statement, is a type of compound statement that consists of two parts, namely an antecedent ([latex]P[/latex]) and a consequent ([latex]Q[/latex]). The implication is often expressed as “If [latex]P[/latex], then [latex]Q[/latex]” or “[latex]P[/latex] implies [latex]Q[/latex]” and is denoted as “[latex]P → Q[/latex]“.

The implication is said to be true except in the case where [latex]P[/latex] is true and [latex]Q[/latex] is false. Here’s a truth table to illustrate:

[latex]P[/latex] [latex]Q[/latex] [latex]P → Q[/latex]
T T T
T F F
F T T
F F T

 

An important concept related to implications is the converse, inverse, and contrapositive:

  • The converse of an implication “[latex]P → Q[/latex]” is “[latex]Q → P[/latex]“. The truth of an implication does not guarantee the truth of its converse. They can be true or false independently of each other.
  • The inverse of an implication “[latex]P → Q[/latex]” is “[latex]¬P → ¬Q[/latex]“. Like the converse, the truth of an implication does not determine the truth of its inverse.
  • The contrapositive of an implication “[latex]P → Q[/latex]” is “[latex]¬Q → ¬P[/latex]“. In logic, an implication and its contrapositive always share the same truth value. If the implication is true, then its contrapositive is true, and vice versa.

Equivalence

In the realm of logic and mathematics, equivalence refers to a relationship between two logical statements where both statements are either true or false simultaneously. This means if we have two statements, [latex]P[/latex] and [latex]Q[/latex], they are considered equivalent, denoted as [latex]P \leftrightarrow Q[/latex], if and only if they are both true or both false.

Let’s illustrate this with a truth table:

[latex]P[/latex] [latex]Q[/latex] [latex]P ↔ Q[/latex]
T T T
T F F
F T F
F F T

 

As the table shows, the biconditional statement ([latex]P \leftrightarrow Q[/latex]) is true only when both [latex]P[/latex] and [latex]Q[/latex] share the same truth value. If [latex]P[/latex] and [latex]Q[/latex] are both true or both false, then [latex]P \leftrightarrow Q[/latex] is true. However, if [latex]P[/latex] and [latex]Q[/latex] have different truth values, then [latex]P \leftrightarrow Q[/latex] is false.

DeMorgan’s Laws

The Main Idea 

In the field of logic and mathematics, DeMorgan’s Laws are fundamental rules that dictate how logical operations of conjunction (AND, represented as [latex]\land[/latex]) and disjunction (OR, represented as [latex]\lor[/latex]) interact with the negation operation (NOT, represented as [latex]\neg[/latex]). DeMorgan’s Laws allow us to simplify complex logical expressions and provide a method for moving a negation operator across a conjunction or disjunction operator.

DeMorgan’s Laws can be stated as follows:

  • [latex]\sim (P \land Q) = \sim P \lor \sim Q[/latex]
  • [latex]\sim (P \lor Q) = \sim P \land \sim Q[/latex]

In plain English, these laws mean:

  • The negation of a conjunction ([latex]P \land Q[/latex]) is equivalent to the disjunction of the negations ([latex]\sim P \lor \sim Q[/latex]).
  • The negation of a disjunction ([latex]P \lor Q[/latex]) is equivalent to the conjunction of the negations ([latex]\sim P \land \sim Q[/latex]).

You can view the transcript for “Logic – DeMorgan’s Laws of Negation” here (opens in new window).