Problem 63
Question
The Statement that is TRUE among the following is (a) The contrapositive of \(3 x+2=8 \Rightarrow x=2\) is \(x \neq 2\) \(\Rightarrow 3 x+2 \neq 8\) (b) The converse of \(\tan x=0 \Rightarrow x=0\) is \(x \neq 0 \Rightarrow \tan x=0\). (c) \(p \Rightarrow q\) is equivalent to \(p \vee \sim q\). (d) \(p \vee q\) and \(p \wedge q\) have the same truth table.
Step-by-Step Solution
Verified Answer
Option (a) is TRUE; it correctly represents the contrapositive.
1Step 1: Understanding Contrapositive
The contrapositive of a statement of form "if \( p \) then \( q \)" is "if not \( q \) then not \( p \)". For the statement \( 3x + 2 = 8 \Rightarrow x = 2 \), the contrapositive is "If \( x eq 2 \) then \( 3x + 2 eq 8 \)." Option (a) is correctly constructed as the contrapositive.
2Step 2: Examining the Converse
The converse of a statement "if \( p \) then \( q \)" is "if \( q \) then \( p \)." The original statement is \( \tan x = 0 \Rightarrow x = 0 \). The converse is "if \( x = 0 \) then \( \tan x = 0 \)," which makes (b) incorrectly constructed.
3Step 3: Evaluating Logical Equivalence
The statement \( p \Rightarrow q \) is logically equivalent to \( \sim p \vee q \), which is different from \( p \vee \sim q \) as mentioned, thus making option (c) false.
4Step 4: Checking Truth Tables
The truth tables for \( p \vee q \) ( logical OR) and \( p \wedge q \) (logical AND) are different, since they represent different operations. Hence, option (d) is false.
Key Concepts
ContrapositiveConverseLogical EquivalenceTruth Tables
Contrapositive
In logic, the contrapositive of a given conditional statement can often provide a way to check the validity of claims. A conditional statement is in the form "if \( p \) then \( q \)," symbolized as \( p \Rightarrow q \). To find its contrapositive, we swap the hypothesis and the conclusion, and negate both. This gives us "if not \( q \) then not \( p \)," or \( \sim q \Rightarrow \sim p \).
For example, the statement \( 3x + 2 = 8 \Rightarrow x = 2 \) has a contrapositive of \( x eq 2 \Rightarrow 3x + 2 eq 8 \). Understanding the contrapositive helps because if the contrapositive is true, then the original statement is also true. However, it's crucial to note that a statement and its contrapositive are logically equivalent, not just converse or opposite. This means proving one true proves the other true as well.
For example, the statement \( 3x + 2 = 8 \Rightarrow x = 2 \) has a contrapositive of \( x eq 2 \Rightarrow 3x + 2 eq 8 \). Understanding the contrapositive helps because if the contrapositive is true, then the original statement is also true. However, it's crucial to note that a statement and its contrapositive are logically equivalent, not just converse or opposite. This means proving one true proves the other true as well.
Converse
The converse of a statement can be easily misunderstood. A converse flips the hypothesis and conclusion. For the statement "if \( p \) then \( q \)," the converse would be "if \( q \) then \( p \)."
In our exercise, the statement \( \tan x = 0 \Rightarrow x = 0 \) is given. Its converse becomes "if \( x = 0 \) then \( \tan x = 0 \)," which is actually quite different from the claim made in the options.
A key point here is that a statement and its converse are not generally logically equivalent. Just because \( p \Rightarrow q \) is true, \( q \Rightarrow p \) isn't necessarily true. Each case should be checked individually to determine validity.
In our exercise, the statement \( \tan x = 0 \Rightarrow x = 0 \) is given. Its converse becomes "if \( x = 0 \) then \( \tan x = 0 \)," which is actually quite different from the claim made in the options.
A key point here is that a statement and its converse are not generally logically equivalent. Just because \( p \Rightarrow q \) is true, \( q \Rightarrow p \) isn't necessarily true. Each case should be checked individually to determine validity.
Logical Equivalence
Logical equivalence refers to scenarios where two statements always have the same truth value. Common logical equivalences help in transforming expressions and solving logical problems.
For example, the statement \( p \Rightarrow q \) is logically equivalent to \( \sim p \vee q \), meaning they return the same truth values in all possible scenarios. In contrast, the condition \( p \vee \sim q \) introduced in the options does not hold equivalence with \( p \Rightarrow q \).
Checking for logical equivalence is important in logic proofs and computer sciences since it often simplifies problems by allowing substitutions of one expression for another without changing the truth value.
For example, the statement \( p \Rightarrow q \) is logically equivalent to \( \sim p \vee q \), meaning they return the same truth values in all possible scenarios. In contrast, the condition \( p \vee \sim q \) introduced in the options does not hold equivalence with \( p \Rightarrow q \).
Checking for logical equivalence is important in logic proofs and computer sciences since it often simplifies problems by allowing substitutions of one expression for another without changing the truth value.
Truth Tables
Truth tables are a fundamental tool for understanding logical operations and statements. They display how truth values change depending on different premises.
For instance, the logical OR operation \( p \vee q \) results in true unless both \( p \) and \( q \) are false, while the logical AND operation \( p \wedge q \) results in true only when both \( p \) and \( q \) are true. Therefore, these operations have different truth tables.
By examining truth tables, one can visibly grasp why two logical statements might be equivalent or not. They play a key role in logic learning, helping someone see the outcome of combinations of truth values for different logical operators.
For instance, the logical OR operation \( p \vee q \) results in true unless both \( p \) and \( q \) are false, while the logical AND operation \( p \wedge q \) results in true only when both \( p \) and \( q \) are true. Therefore, these operations have different truth tables.
By examining truth tables, one can visibly grasp why two logical statements might be equivalent or not. They play a key role in logic learning, helping someone see the outcome of combinations of truth values for different logical operators.
Other exercises in this chapter
Problem 61
The negation of the statement "If I become a teacher, then I will open a school", is: (a) I will become a teacher and I will not open a school. (b) Either I wil
View solution Problem 62
Let \(p\) and \(q\) denote the following statements \(p:\) The sun is shining \(q\) : I shall play tennis in the afternoon The negation of the statement "If the
View solution Problem 64
Let \(\mathrm{S}\) be a non-empty subset of \(\mathrm{R}\). Consider the following statement : \(P\) : There is a rational number \(x \in S\) such that \(x>0 .\
View solution Problem 60
The contrapositive of the statement "I go to school if it does not rain" is (a) If it rains, I do not go to school. (b) If I do not go to school, it rains. (c)
View solution