Problem 41 - Thinking Mathematically

Question

Determine which, if any, of the three given statements are equivalent. You may use information about a conditional statement's converse, inverse, or contrapositive, De Morgan's laws, or truth tables. a. It is not true that I have a ticket and cannot go. b. I do not have a ticket and can go. c. I have a ticket or I cannot go.

Step-by-Step Solution

Answer

None of the provided statements are equivalent.

Step 1: Express the Statements in Logical Form

Let's denote 'I have a ticket' as \(A\) and 'I can go' as \(B\). We can then express the statements as follows: \n a. \(\neg(A \, \land \, \neg B)\) \n b. \(\neg A \, \land \, B\) \n c. \(A \, \lor \, \neg B\)

Step 2: Simplify the Statements Using De Morgan's Laws

De Morgan's Laws state that: \n 1) The negation of a conjunction is the disjunction of the negations. \n 2) The negation of a disjunction is the conjunction of the negations. \n Applying De Morgan's Law to statement a, we get: \(\neg A \, \lor \, B \)

Step 3: Compare the Statements

Comparing the simplified form of statement a to the others: \n Statement a: \(\neg A \, \lor \, B\) (It is not true that I do not have a ticket, or I can go) is not equivalent to any of the other statements \n Statement b: \(\neg A \, \land \, B\) (I do not have a ticket, and I can go) \n Statement c: \(A \, \lor \, \neg B\) (I have a ticket, or I can not go)

Key Concepts

De Morgan's Laws

Understanding De Morgan's Laws is like learning a shortcut that can simplify complex logical statements. These laws, named after mathematician Augustus De Morgan, show us how to handle the negation of conjunctions (and statements) and disjunctions (or statements).

There are two main principles in De Morgan's Laws:

• The negation of a conjunction \( A \land B \) is the disjunction of the negations \( eg A \lor eg B \).
• The negation of a disjunction \( A \lor B \) is the conjunction of the negations \( eg A \land eg B \).

This means you can flip and 'negate' the terms inside and change the connector. For example, saying 'not both A and B' is the same as saying 'either not A or not B'. Applying this to our exercise simplified statement a, turning it from 'not (A and not B)' into 'not A or B', which can then be compared to other statements to check for equivalency. Using De Morgan's laws turns out to be indispensable for logical reasoning and computer science applications.

Truth Tables

Let's talk about truth tables, a fundamental tool for anyone dabbling with logical statements. They help visualize how different logic combinations interact. Essentially, it is like a ledger listing every possible truth value for the components of a logical statement and showing the outcome.

Creating a truth table is straightforward:

• Write down all possible combinations of truth values for the given variables.
• Calculate the resulting truth value of the logical expression for each combination.

These tables are super handy when checking for equivalence between statements. If two statements have the same truth values under all possible scenarios, they are equivalent. In our exercise, we could create a truth table for each statement and check if the outputs match to determine equivalence. However, we opted for De Morgan's laws, a quicker route in this case.

Logical Negation

Logical negation is the flip side of a statement, like saying 'no' instead of 'yes' or 'false' when it's actually 'true'. It's the 'not' in logic. When we negate a statement, denoted by \( eg \), we are effectively stating the opposite. For instance, if 'A' is 'I am hungry', \( eg A \) would be 'I am not hungry'.

Negation is crucial because it allows us to express what is not true, which is just as important as stating what is true. In conditional logic, the negation often shifts the outcome of an expression completely. For example, \( eg(A \land B) \) doesn't just mean 'not A and B', but rather, after applying De Morgan's, it means 'not A or not B'. In our exercise, we used logical negation to interpret statement a and transform it using De Morgan's law for comparison with the other statements.

Converse Inverse Contrapositive

These fancy terms actually represent simple variations of a conditional statement. Starting with a basic if-then statement 'If A, then B', here's what these variations mean:

• The Converse flips the hypothesis and conclusion: 'If B, then A'.
• The Inverse negates both the hypothesis and conclusion: 'If not A, then not B'.
• The Contrapositive negates and flips both: 'If not B, then not A'.

Now, here comes the interesting part. The contrapositive is always logically equivalent to the original statement, whereas the converse and inverse are not. This fact is super useful in proving logical equivalences or solving riddles. When we assess statements, like in our exercise, understanding these relations can help determine the equivalency or lack thereof between different statements.