Construct a truth table for each of these compound propositions. a) p ∧p b) p ∨p c) (p ∨q) →q d)

Construct a truth table for each of these compound...

Key Concepts

Contradiction

A contradiction is a compound proposition that is invariably false, no matter what truth values are assigned to its individual components. Recognizing contradictions is key in logical analysis as they indicate impossibility.

Tautology

A tautology is a compound proposition that is always true regardless of the individual truth values of its components. It reveals a universal truth within the logical framework.

Biconditional

The biconditional operator, denoted by ‘?’, asserts that two propositions are equivalent, being true exactly when both have the same truth value. It is useful for establishing logical equivalence between statements.

Implication

Implication, often expressed as ‘?’, is a logical operator that represents a conditional statement. It is false only when the antecedent is true and the consequent is false, encapsulating the idea that a condition leads to a result.

Negation

Negation is an operator denoted by ‘¬’ that inverts the truth value of a proposition, turning true into false and vice versa. It is essential for constructing opposites or complementing conditions in logic.

Disjunction

Disjunction, represented by ‘?’, is a logical operator that results in true when at least one of its operands is true. It captures the logical ‘or’ connection in compound propositions.

Logical Connectives

Logical connectives are operators used to build compound propositions. They include conjunction (and), disjunction (or), negation (not), implication (if...then), and biconditional (if and only if), each having a specific way of combining or modifying truth values.

Compound Proposition

A compound proposition is a statement formed by combining simpler statements using logical operators. Analyzing its structure helps in determining its overall truth value under various conditions.

Truth Table

A truth table is a systematic way to list all possible truth values of a set of logical variables and then compute the truth value of a compound proposition for each combination. It is fundamental for understanding logical equivalences and the behavior of logical operations.

Conjunction

Conjunction is a logical operator, typically represented by ‘?’, which results in true if and only if both of its operands are true. It is used to represent the ‘and’ relationship between statements.

Transcript

00:01 Okay, so for this question we have a series of compound propositions and we want to construct the truth tables for them so for the first one we have p and not p so to construct this really these columns are kind of arbitrary you don't really need this not p if you can do it in your brain yourself but typically what people do when we construct these truth tables is right every single single little thing that we can.

00:32 So for this case, we have one row, one column for p and one column for not p.

00:38 So we have p would be true and false and then not p will be the opposite of p.

00:44 So opposite of true would be false, opposite of false would be true.

00:47 And then this one will be true and false.

00:50 So that's just false.

00:52 False and true, that's false again.

00:54 So for this question we have p or not p.

00:58 So once again we will we will fill out p and we fill out not p.

01:02 So first we have to ignore these two rows and just fill out p first which is just true and false.

01:09 Not p is just false, not false is true.

01:12 So this will be true or false.

01:14 So that's true.

01:15 This is false or true.

01:17 That's true.

01:19 Now for this question we have p or not q implies p.

01:26 So for this question what we'll do is we need to construct one column of p, one column of q.

01:32 So these will be the base variables more or less.

01:36 Then because we have a not q, we typically just introduce this not q.

01:41 Then we'll do this whole thing.

01:44 So this is p or not q.

01:47 And then finally we'll do the entire thing, which is p or not q implies p.

01:55 So let's just run, let's just do row by row.

02:01 So yeah, row...

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