Problem 43 - 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. If the grass turns yellow, you did not use fertilizer or water. b. If you use fertilizer and water, the grass will not turn yellow. c. If the grass does not turn yellow, you used fertilizer and water.

Step-by-Step Solution

Answer

Statements a and b are equivalent and statement c is not equivalent to a or b.

Step 1: Analyzing Statement a

The given statement a 'If the grass turns yellow, you did not use fertilizer or water.' is a conditional statement. We can denote it as ‘If P, then Q’, where P is 'the grass turns yellow' and Q is 'you did not use fertilizer or water'.

Step 2: Analyzing Statement b

The given statement b 'If you use fertilizer and water, the grass will not turn yellow.' is also a conditional statement. We can denote it as 'If not Q, then not P', where 'not Q' is 'using fertilizer and water' and 'not P' is 'the grass not turning yellow'.

Step 3: Analyzing Statement c

Statement c 'If the grass does not turn yellow, you used fertilizer and water.' is another conditional statement, which can be denoted as 'If not P, then not Q', where 'not P' is 'the grass does not turn yellow' and 'not Q' is 'you did use fertilizer and water'.

Step 4: Checking Equivalence

From the analysis, we can see that statement a and statement b are equivalent as they are the contrapositive of each other. Both are following the format 'If P, then Q' and 'If not Q, then not P'. However, statement c is not equivalent to statement a or b because the conditions do not match.

Key Concepts

Conditional Statements

Conditional statements are statements that contain a hypothesis followed by a conclusion. They have the structure 'If P, then Q', where P is the hypothesis and Q is the conclusion. For example, "If the grass turns yellow, you did not use fertilizer or water." is a conditional statement where P is "the grass turns yellow" and Q is "you did not use fertilizer or water."

Conditional statements are important for logic reasoning as they help establish relationships between statements. They are the backbone of logical argumentation and are frequently used in mathematics, computer science, and everyday reasoning. It's crucial to understand the format and to identify P and Q properly.

Understanding conditional statements allows us to rewrite them into equivalent forms, which will be useful when we dissect concepts like 'contrapositives' later in this article.

Truth Tables

Truth tables are a great tool for determining the truth value of logical expressions. A truth table shows all possible truth values for its components, which help us analyze the logical equivalence between different statements.

In the case of the statements given, a truth table would include four rows, representing combinations of truth values (True or False) for the propositions involved. For instance, two propositions, P and Q, would have a truth table like this:

• P: True, Q: True
• P: True, Q: False
• P: False, Q: True
• P: False, Q: False

The table would then show the result of the conditional statement 'If P, then Q' for each combination. For conditional statements, the statement is only false when P is true and Q is false.

Using truth tables can help to visually verify the equivalence between different logical statements, demonstrating clearly why certain statements (like a and b in this exercise) are equivalent.

Contrapositives

Contrapositives are a transformation of the original conditional statement that maintains logical equivalence. The contrapositive of an 'If P, then Q' statement is 'If not Q, then not P'.

This transformation is crucial in logical reasoning because the contrapositive of a statement is always logically equivalent to the original condition. This means if the original statement is true, its contrapositive is also true.

In our exercise, statements a and b represent a conditional statement and its contrapositive, respectively, thereby confirming their logical equivalence:

• Statement a: 'If P, then Q'
• Statement b: 'If not Q, then not P'

Recognizing the contrapositive helps in proving logical statements and understanding relations, especially when conditions appear reversed or inverted.

De Morgan's Laws

De Morgan's Laws provide rules for the negation of logical expressions, specifically for handling 'and' (\(\land\)) and 'or' (\(\lor\)) operators. These laws are tremendously useful in simplifying logical expressions and proving equivalences.

De Morgan's Laws states:

• Not (P and Q) is equivalent to (Not P) or (Not Q)
• Not (P or Q) is equivalent to (Not P) and (Not Q)

For example, when applying these laws, De Morgan's Laws helped analyze the logical structure needed in statement a in the original exercise. Here the Q part "you did not use fertilizer or water" can be broken down using De Morgan's, ensuring a better understanding of the logical form.

These transformations are critical when dealing with complex logical arguments or proofs, as they provide a clear pathway to reinterpreting statements and simplifying them.