Use truth tables to verify the associative laws a) (p ∨q) ∨r ≡p ∨(q ∨r) b) (p ∧q) ∧r ≡p ∧(q ∧r)

Name: Problem 4, Chapter 1: Discrete Mathematics and its Applications
Uploaded: 2019-09-17T19:25:53-08:00
Duration: 5 min 37 s
Channel: Dalia Rodriguez
Description: Use truth tables to verify the associative laws a) (p ∨q) ∨r ≡p ∨(q ∨r) b) (p ∧q) ∧r ≡p ∧(q ∧r)

step 1 Okay, so we want to verify the associated law. We will begin by clarifying that the left side is

Use truth tables to verify the associative laws a) (p ∨q) ∨r...

Key Concepts

Truth Tables

Truth tables are systematic tabulations used to determine the truth value of logical expressions under every possible combination of truth values for their atomic components. They provide a clear methodology for verifying logical equivalences, contradictions, or tautologies by enumerating all possible scenarios and checking the outcome of the propositions.

Associative Law

The associative law in logic states that when performing the same binary operation (such as disjunction or conjunction) on multiple operands, the grouping of those operands does not affect the final result. In other words, the expressions (A ? B) ? C and A ? (B ? C) (and similarly for conjunction) are logically equivalent, which allows for flexibility in rewriting and simplifying logical statements.

Logical Disjunction

Logical disjunction is a basic logical operation, often symbolized as ?, that represents 'or' between two propositions. The disjunction A ? B is true if at least one of the propositions A or B is true. This operation is key to understanding various logical expressions, including applications of the associative law.

Logical Conjunction

Logical conjunction is another fundamental logical operation, symbolized as ?, that represents 'and' between two propositions. The conjunction A ? B is true only when both A and B are true. Like disjunction, conjunction plays an important role in the structure of logical formulas and the application of associative rules.

Logical Equivalence

Logical equivalence, denoted by the symbol ?, indicates that two logical expressions have the same truth value across all possible truth assignments to their components. Verifying logical equivalence using truth tables is a common method in logic to show that different formulations of expressions yield identical results under every possible evaluation.

Transcript

00:01 Okay, so we want to verify the associated law.

00:05 We will begin by clarifying that the left side is equivalent to the right side of the associate of the by using a truth table.

00:17 If the left side and the right side have the same truth table, then that means that they are logically equivalent.

00:23 So i have began by writing all the possible combinations of truth values for pq and r.

00:29 Now we will begin by doing the rest.

00:32 So p or q is true if either p or q or both are true.

00:37 Right here it's true, true, true, true, false, since both p and q are false.

00:46 Again, true, since p is true, true, true, and false, true.

01:06 Now, p or q is true if either p or q or r or both are true.

01:12 So, p or q is true for all.

01:14 Of these and that automatically makes p or q or r true.

01:20 Now, p or q is false here but r is true, so this is true and here p or q is false and r is also false.

01:30 So this is false.

01:32 Q or r is true if q is true or both q and r are true.

01:39 This is true, this is true, this is true.

01:43 Here q is true, both q and r are true.

01:50 R is true, both q and r fall here, and q is true here, and both q and r falls here.

02:01 So this is false.

02:03 Now p or qr is true if p is true or q or r is true, so qr is true for all these first five.

02:11 So let's just automatically put these as true.

02:18 And then here qr is true, but p is true.

02:22 True so this is true here qr r is true again so this is true and here qr r is false and also r is false so this is false we can observe that p or q or r has the same truth table as p or q or r so therefore they are logically equivalent now we will do the second part which is p and q and r is logically equivalent to p and q and r.

02:57 So once again i have filled in all the possible combinations of truth values for p q and r and we will do the rest.

03:07 So p and q is true only when p and q are both true.

03:11 Otherwise it's false...

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