Related Conditional Statements
Converse of a Conditional Statement
Given an original conditional statement (implication) of the form "If $p$, then $q$" (symbolically, $p \implies q$), which asserts that the truth of $p$ leads to the truth of $q$, we can construct other related conditional statements by rearranging or modifying its components. The first of these related statements is called the converse.
Definition of the Converse
The converse of a conditional statement $p \implies q$ is the new conditional statement formed by interchanging the hypothesis ($p$) and the conclusion ($q$). The direction of the implication is reversed.
- Original Statement: $p \implies q$ (Read as: "If $p$, then $q$")
- Converse Statement: $q \implies p$ (Read as: "If $q$, then $p$")
Essentially, the condition in the original statement becomes the consequence in the converse, and the consequence in the original becomes the condition in the converse.
Truth Value Relationship: Are they Logically Equivalent?
A very important point to grasp is that a conditional statement and its converse are not necessarily logically equivalent. This means that the truth value of the original statement $p \implies q$ is not always the same as the truth value of its converse $q \implies p$. You cannot assume that just because an "if-then" statement is true, its reverse is also true.
We can demonstrate this non-equivalence by comparing their truth tables. For statements involving two variables $p$ and $q$, there are $2^2 = 4$ possible combinations of truth values:
| $p$ | $q$ | $p \implies q$ | $q \implies p$ (Converse) |
|---|---|---|---|
| T | T | T | T |
| T | F | F | T |
| F | T | T | F |
| F | F | T | T |
Looking at the columns for $p \implies q$ and $q \implies p$, we can see that their truth values differ in the second row (where $p$ is True and $q$ is False, $p \implies q$ is False while $q \implies p$ is True) and in the third row (where $p$ is False and $q$ is True, $p \implies q$ is True while $q \implies p$ is False). Since their truth values are not identical for all possible assignments of truth values to $p$ and $q$, $p \implies q$ and $q \implies p$ are not logically equivalent ($p \implies q \not\equiv q \implies p$).
This table confirms that:
- It is possible for a true statement to have a true converse (e.g., when $p$ and $q$ are both True, or both False).
- It is possible for a false statement to have a true converse (when $p$ is True and $q$ is False).
- It is possible for a true statement to have a false converse (when $p$ is False and $q$ is True).
Example 1. Write the converse of the statement: "If a number is divisible by 4, then it is divisible by 2." Determine the truth value of both the original statement and its converse.
Answer:
Let $p$: A number is divisible by 4.
Let $q$: A number is divisible by 2.
Original Statement ($p \implies q$): "If a number is divisible by 4, then it is divisible by 2."
Evaluation:
Consider any number that is divisible by 4, for example, 8. $8 = 4 \times 2$. Since 4 is $2 \times 2$, we can write $8 = (2 \times 2) \times 2 = 2 \times (2 \times 2) = 2 \times 4$. So, 8 is also divisible by 2. This applies to any number divisible by 4: if a number $N$ is divisible by 4, then $N = 4k$ for some integer $k$. We can rewrite this as $N = 2 \times (2k)$. Since $2k$ is an integer, $N$ is also divisible by 2. Thus, if the hypothesis ($p$) is true, the conclusion ($q$) is always true. This means the original statement is universally True.
Truth Value of Original Statement: True (T)
Converse Statement ($q \implies p$): Formed by swapping $p$ and $q$. "If a number is divisible by 2, then it is divisible by 4."
Evaluation:
To check the truth value of this statement, let's try to find a counterexample – a number that is divisible by 2 (making the hypothesis true) but not divisible by 4 (making the conclusion false). Consider the number 6. Is 6 divisible by 2? Yes, $6 = 2 \times 3$. So, the hypothesis "a number is divisible by 2" ($q$) is True for the number 6. Is 6 divisible by 4? No, 6 divided by 4 is 1 with a remainder of 2. So, the conclusion "it is divisible by 4" ($p$) is False for the number 6. We have found a case where $q$ is True and $p$ is False (T $\implies$ F). According to the truth table for implication, a conditional statement with a True hypothesis and a False conclusion is False.
Truth Value of Converse Statement: False (F)
This example clearly illustrates that a true conditional statement can have a false converse, demonstrating that they are not logically equivalent.
Example 2. Write the converse of the statement: "If a triangle is equilateral, then it is isosceles." Determine the truth value of both statements.
Answer:
Let $p$: A triangle is equilateral (all three sides are equal).
Let $q$: A triangle is isosceles (at least two sides are equal).
Original Statement ($p \implies q$): "If a triangle is equilateral, then it is isosceles."
Evaluation:
By definition, an equilateral triangle is one in which all three sides are of equal length. If a triangle has three equal sides, then it certainly has at least two equal sides. Thus, the condition of being equilateral ($p$) guarantees the condition of being isosceles ($q$). If $p$ is true, $q$ must be true. This statement is True.
Truth Value of Original Statement: True (T)
Converse Statement ($q \implies p$): Formed by swapping $p$ and $q$. "If a triangle is isosceles, then it is equilateral."
Evaluation:
If a triangle is isosceles, meaning it has at least two equal sides, does it necessarily mean all three sides are equal (i.e., it is equilateral)? No. Consider a triangle with side lengths 5 cm, 5 cm, and 8 cm. This is an isosceles triangle (so $q$ is True) because two sides are equal. However, it is not an equilateral triangle (so $p$ is False) because the third side is different. This is a case of T $\implies$ F, making the converse False.
Truth Value of Converse Statement: False (F)
Again, a true statement can have a false converse, further demonstrating that they are not logically equivalent.
Competitive Exam Pointer: Converse
Questions about related conditional statements, especially the converse, are very common. Key points for competitive exams:
- Formation: The converse of $p \implies q$ is simply $q \implies p$. Swap the hypothesis and conclusion.
- Logical Equivalence: **Crucially, $p \implies q$ and its converse $q \implies p$ are NOT logically equivalent.**
- Truth Values: Be prepared to evaluate the truth value of the converse independently of the original statement. You may need to provide counterexamples (like the number 6 for divisibility, or a non-square rectangle, or an isosceles triangle that is not equilateral) to show a converse is false.
- Terminology: Be familiar with the term "converse".
Understanding this difference is fundamental to avoiding logical errors in arguments and proofs.
Inverse of a Conditional Statement
The second related conditional statement derived from an original implication $p \implies q$ is the inverse. While the converse involves swapping the parts of the implication, the inverse involves negating them while keeping the original direction of implication.
Definition of the Inverse
Given a conditional statement $p \implies q$, its inverse is the conditional statement formed by negating both the hypothesis ($p$) and the conclusion ($q$).
- Original Statement: $p \implies q$ (If $p$, then $q$)
- Inverse Statement: $\sim p \implies \sim q$ (If not $p$, then not $q$)
The inverse statement essentially says: "If the original condition does not hold ($\sim p$), then the original conclusion does not hold either ($\sim q$)."
Truth Value Relationship: Are they Logically Equivalent?
Similar to the converse, the inverse of a conditional statement is **not** logically equivalent to the original statement ($p \implies q$). Their truth values do not necessarily match for all cases of $p$ and $q$. You cannot assume that if an "if-then" statement is true, its "if not-then not" form is also true.
Let's compare their truth values using truth tables:
| $p$ | $q$ | $\sim p$ | $\sim q$ | $p \implies q$ | $\sim p \implies \sim q$ (Inverse) |
|---|---|---|---|---|---|
| T | T | F | F | T $\implies$ T = T | F $\implies$ F = T |
| T | F | F | T | T $\implies$ F = F | F $\implies$ T = T |
| F | T | T | F | F $\implies$ T = T | T $\implies$ F = F |
| F | F | T | T | F $\implies$ F = T | T $\implies$ T = T |
Looking at the columns for $p \implies q$ and $\sim p \implies \sim q$, we see they differ in the second row (where $p$ is True and $q$ is False, $p \implies q$ is False while $\sim p \implies \sim q$ is True) and in the third row (where $p$ is False and $q$ is True, $p \implies q$ is True while $\sim p \implies \sim q$ is False). Therefore, $p \implies q$ and its inverse $\sim p \implies \sim q$ are not logically equivalent ($p \implies q \not\equiv \sim p \implies \sim q$).
An important relationship exists between the inverse and the converse: The inverse statement ($\sim p \implies \sim q$) is logically equivalent to the converse statement ($q \implies p$). This can be seen by comparing the last column of the table above ($\sim p \implies \sim q$) with the last column of the truth table for Converse ($q \implies p$) in the previous section. Their truth values are identical (T, T, F, T). This logical equivalence, $q \implies p \equiv \sim p \implies \sim q$, will be formally discussed and proven using truth tables in the next section (I4 of this H1 tag).
Example 1. Write the inverse of the statement: "If a number is divisible by 4, then it is divisible by 2." Determine the truth value of both statements.
Answer:
Let $p$: A number is divisible by 4.
Let $q$: A number is divisible by 2.
The negations are:
$\sim p$: A number is not divisible by 4.
$\sim q$: A number is not divisible by 2.
Original Statement ($p \implies q$): "If a number is divisible by 4, then it is divisible by 2."
Evaluation:
As shown in Example 1 of the Converse section, this statement is universally True.
Truth Value of Original Statement: True (T)
Inverse Statement ($\sim p \implies \sim q$): "If a number is not divisible by 4, then it is not divisible by 2."
Evaluation:
To check the truth value of this statement, let's try to find a counterexample – a number for which the hypothesis "a number is not divisible by 4" ($\sim p$) is True, but the conclusion "it is not divisible by 2" ($\sim q$) is False. Consider the number 6. Is 6 not divisible by 4? Yes, it is not. So, $\sim p$ is True for 6. Is 6 not divisible by 2? No, it is divisible by 2. So, $\sim q$ is False for 6. We have found a case where the hypothesis ($\sim p$) is True and the conclusion ($\sim q$) is False (T $\implies$ F). According to the truth table for implication, T $\implies$ F is False.
Truth Value of Inverse Statement: False (F)
This example shows that a true conditional statement can have a false inverse, confirming they are not logically equivalent.
Example 2. Write the inverse of the statement: "If a triangle is equilateral, then it is equiangular." Determine the truth value of both statements.
Answer:
Let $p$: A triangle is equilateral.
Let $q$: A triangle is equiangular.
The negations are:
$\sim p$: A triangle is not equilateral.
$\sim q$: A triangle is not equiangular.
Original Statement ($p \implies q$): "If a triangle is equilateral, then it is equiangular."
Evaluation:
It is a fundamental theorem in geometry that a triangle with three equal sides must also have three equal angles. Thus, if $p$ is true, $q$ is always true. This statement is universally True.
Truth Value of Original Statement: True (T)
Inverse Statement ($\sim p \implies \sim q$): "If a triangle is not equilateral, then it is not equiangular."
Evaluation:
Is this statement always true? Yes. A triangle is equiangular if and only if it is equilateral. This means that being "not equilateral" ($\sim p$) is a condition that holds precisely when being "not equiangular" ($\sim q$) holds. Thus, $\sim p$ is True if and only if $\sim q$ is True. This corresponds to the (T $\implies$ T) and (F $\implies$ F) cases in the truth table for implication. So, if $\sim p$ is True, $\sim q$ must be True (T $\implies$ T is True). If $\sim p$ is False (meaning $p$ is true, hence $q$ is true), then $\sim q$ is also False (F $\implies$ F is True). In all cases, the inverse statement is True.
Truth Value of Inverse Statement: True (T)
In this specific example, both the original statement and its inverse happen to be true. However, this is not a general rule, as shown by Example 1 and the truth table comparison. They are not logically equivalent because their truth values do not match in all possible logical scenarios (specifically, rows 2 and 3 in their truth table comparison). This example corresponds only to rows 1 and 4 of the comparison table, where their truth values coincide.
Competitive Exam Pointer: Inverse
Understanding the inverse is key for questions involving related conditional statements and logical equivalences.
- Formation: The inverse of $p \implies q$ is $\sim p \implies \sim q$. Negate both the hypothesis and the conclusion, keeping the implication direction.
- Logical Equivalence: **$p \implies q$ and its inverse $\sim p \implies \sim q$ are NOT logically equivalent.** Their truth values can differ depending on the truth values of $p$ and $q$.
- Relationship to Converse: The inverse ($\sim p \implies \sim q$) IS logically equivalent to the converse ($q \implies p$). This is a very important equivalence to remember and can be verified using truth tables.
- Truth Values: Be prepared to evaluate the truth value of the inverse. Like the converse, it doesn't automatically inherit the truth value of the original statement.
Incorrectly assuming the inverse is equivalent to the original statement is a common source of logical errors.
Contrapositive of a Conditional Statement
In addition to the converse and inverse, there is a third very important related conditional statement derived from an original implication $p \implies q$. This statement is called the contrapositive, and it holds a special logical relationship with the original statement.
Definition of the Contrapositive
Given a conditional statement $p \implies q$ ("If $p$, then $q$"), its contrapositive is the conditional statement formed by interchanging (swapping) the hypothesis ($p$) and the conclusion ($q$) **and** negating both of them.
You can think of the formation process in a couple of ways, both leading to the same result:
- Start with the original statement: $p \implies q$
- Negate both the hypothesis and the conclusion: $\sim p$ and $\sim q$.
- Form a conditional statement with the *negated conclusion* as the new hypothesis and the *negated hypothesis* as the new conclusion: $\sim q \implies \sim p$.
Alternatively, you could first form the converse and then negate both parts:
- Start with the original statement: $p \implies q$
- Form the converse by swapping: $q \implies p$
- Negate both the hypothesis ($q$) and the conclusion ($p$) of the converse: $\sim q \implies \sim p$.
Both methods lead to the definition:
- Original Statement: $p \implies q$ (If $p$, then $q$)
- Contrapositive Statement: $\sim q \implies \sim p$ (If not $q$, then not $p$)
The contrapositive asserts: "If the conclusion ($q$) does not happen, then the original condition ($p$) could not have happened either."
Truth Value Relationship: Logical Equivalence!
This is the most significant and useful relationship among the related conditional statements: A conditional statement and its contrapositive are **logically equivalent**. They always have the same truth value, regardless of the specific statements $p$ and $q$ and their truth values. This means $p \implies q \equiv \sim q \implies \sim p$.
The logical equivalence between an implication and its contrapositive is a fundamental principle in logic and mathematics. It is the basis for a powerful proof technique called **Proof by Contrapositive**, where to prove "If $p$, then $q$" is true, one instead proves its contrapositive "If not $q$, then not $p$" is true.
We can formally verify this logical equivalence using a truth table. We need columns for $p, q$, their negations $\sim p, \sim q$, the original statement $p \implies q$, and the contrapositive $\sim q \implies \sim p$.
| $p$ | $q$ | $\sim p$ | $\sim q$ | $p \implies q$ | $\sim q \implies \sim p$ |
|---|---|---|---|---|---|
| T | T | F | F | T $\implies$ T = T | F $\implies$ F = T |
| T | F | F | T | T $\implies$ F = F | T $\implies$ F = F |
| F | T | T | F | F $\implies$ T = T | F $\implies$ T = T |
| F | F | T | T | F $\implies$ F = T | T $\implies$ T = T |
Comparing the column for the original statement ($p \implies q$) and the column for the contrapositive ($\sim q \implies \sim p$, the last column), we see that the truth values are identical in every single row (T, F, T, T). This identity in truth values across all possible scenarios formally proves that $p \implies q$ is logically equivalent to $\sim q \implies \sim p$.
Example 1. Write the contrapositive of the statement: "If a number is divisible by 4, then it is divisible by 2." Determine the truth value of both statements.
Answer:
Let $p$: A number is divisible by 4.
Let $q$: A number is divisible by 2.
The negations are:
$\sim p$: A number is not divisible by 4.
$\sim q$: A number is not divisible by 2.
Original Statement ($p \implies q$): "If a number is divisible by 4, then it is divisible by 2."
Evaluation:
As shown in Example 1 of the Converse section, this statement is universally True. Any multiple of 4 is indeed a multiple of 2.
Truth Value of Original Statement: True (T)
Contrapositive Statement ($\sim q \implies \sim p$): Formed by swapping and negating $p$ and $q$. "If a number is not divisible by 2, then it is not divisible by 4."
Evaluation:
Let's check the truth value of the contrapositive. The hypothesis is "a number is not divisible by 2" ($\sim q$). If a number is not divisible by 2, it means the number is odd. The conclusion is "it is not divisible by 4" ($\sim p$). Can an odd number be divisible by 4? No, because any number divisible by 4 must be even ($4 \times k = 2 \times (2k)$). Therefore, if the hypothesis ($\sim q$, number is odd) is True, the conclusion ($\sim p$, number is not divisible by 4) must also be True (T $\implies$ T is True). What about when the hypothesis ($\sim q$) is False? This means the number *is* divisible by 2 (i.e., even). In this case ($\sim q$ is False), the implication $\sim q \implies \sim p$ is automatically True, regardless of whether the number is divisible by 4 or not (F $\implies$ anything is True). Thus, the contrapositive statement is always True.
Truth Value of Contrapositive Statement: True (T)
As expected from the logical equivalence, the original statement and its contrapositive have the same truth value (both True).
Example 2. Write the contrapositive of the statement: "If a triangle is equilateral, then it is equiangular." Determine the truth value of both statements.
Answer:
Let $p$: A triangle is equilateral.
Let $q$: A triangle is equiangular.
The negations are:
$\sim p$: A triangle is not equilateral.
$\sim q$: A triangle is not equiangular.
Original Statement ($p \implies q$): "If a triangle is equilateral, then it is equiangular."
Evaluation:
It is a fundamental theorem in geometry that if a triangle has three equal sides (equilateral), then it also has three equal angles (equiangular). Thus, this statement is universally True.
Truth Value of Original Statement: True (T)
Contrapositive Statement ($\sim q \implies \sim p$): "If a triangle is not equiangular, then it is not equilateral."
Evaluation:
If a triangle is not equiangular ($\sim q$ is True), it means its angles are not all equal. By a geometric theorem, a triangle is equiangular if and only if it is equilateral. This implies that if a triangle is not equiangular, it cannot be equilateral ($\sim p$ is True). So, if the hypothesis ($\sim q$) is True, the conclusion ($\sim p$) must be True (T $\implies$ T is True). If the hypothesis ($\sim q$) is False (meaning the triangle *is* equiangular, $q$ is True), then the implication $\sim q \implies \sim p$ is True regardless of $\sim p$ (F $\implies$ anything is True). Thus, the contrapositive statement is always True.
Truth Value of Contrapositive Statement: True (T)
Again, the original statement and its contrapositive have the same truth value (both True), as expected from their logical equivalence.
Competitive Exam Pointer: Contrapositive
The contrapositive is arguably the most important related conditional statement for competitive exams, especially in topics related to proofs and logical equivalence.
- Formation: The contrapositive of $p \implies q$ is $\sim q \implies \sim p$. Swap the hypothesis and conclusion AND negate both.
- Logical Equivalence: **A conditional statement ($p \implies q$) is ALWAYS logically equivalent to its contrapositive ($\sim q \implies \sim p$).** They have the same truth value in all possible scenarios.
- Usage in Proofs: The equivalence $p \implies q \equiv \sim q \implies \sim p$ means that to prove "If $p$, then $q$", you can instead prove "If not $q$, then not $p$". This is the method of **Proof by Contrapositive**.
- Relationship to Inverse/Converse: The contrapositive ($\sim q \implies \sim p$) is logically equivalent to the converse ($q \implies p$) AND the inverse ($\sim p \implies \sim q$). (This is incorrect, the contrapositive is only equivalent to the original, and converse is equivalent to inverse. Let me correct this point in the exam pointer). Let's rephrase: The contrapositive is equivalent to the inverse of the converse, and the converse is equivalent to the inverse of the contrapositive. But the key pairs are (Original, Contrapositive) and (Converse, Inverse).
Masterise the formation and, critically, the logical equivalence of the contrapositive to the original statement.
Relationships and Logical Equivalence among Related Conditional Statements
We have defined the three main conditional statements related to an original conditional statement $p \implies q$:
- Original Statement: $p \implies q$ (If $p$, then $q$)
- Converse: $q \implies p$ (If $q$, then $p$)
- Inverse: $\sim p \implies \sim q$ (If not $p$, then not $q$)
- Contrapositive: $\sim q \implies \sim p$ (If not $q$, then not $p$)
We have already seen through examples and truth table comparisons that the original statement is generally not equivalent to its converse or inverse. However, there are specific and very important logical equivalences among these four statements.
Key Logical Equivalences
Based on the definitions and truth table evaluations, we can formally state the key logical equivalences:
- Equivalence 1: The Original statement ($p \implies q$) is logically equivalent to its Contrapositive ($\sim q \implies \sim p$). Symbolically: $p \implies q \equiv \sim q \implies \sim p$.
- Equivalence 2: The Converse statement ($q \implies p$) is logically equivalent to the Inverse statement ($\sim p \implies \sim q$). Symbolically: $q \implies p \equiv \sim p \implies \sim q$.
These equivalences mean that within a logical argument or proof, you can replace a statement with its logically equivalent form without changing the truth value of the overall structure. If one statement in an equivalence pair is true, the other is also true. If one is false, the other is also false.
Conversely, the lack of equivalence between the Original and Converse ($p \implies q \not\equiv q \implies p$) and the Original and Inverse ($p \implies q \not\equiv \sim p \implies \sim q$) means you cannot swap these freely in logic without potentially changing the truth of the argument.
Truth Table Demonstration of All Relationships
To see all these relationships clearly, let's construct a single truth table that includes all four related statements and their truth values for every possible combination of $p$ and $q$.
| $p$ | $q$ | $\sim p$ | $\sim q$ | Original ($p \implies q$) |
Converse ($q \implies p$) |
Inverse ($\sim p \implies \sim q$) |
Contrapositive ($\sim q \implies \sim p$) |
|---|---|---|---|---|---|---|---|
| T | T | F | F | T | T | T | T |
| T | F | F | T | F | T | T | F |
| F | T | T | F | T | F | F | T |
| F | F | T | T | T | T | T | T |
From this comprehensive table, we can visually confirm the equivalences:
- Compare the column for "Original" ($p \implies q$) with the column for "Contrapositive" ($\sim q \implies \sim p$). They are identical in every row, highlighted in light green. This proves $p \implies q \equiv \sim q \implies \sim p$.
- Compare the column for "Converse" ($q \implies p$) with the column for "Inverse" ($\sim p \implies \sim q$). They are identical in every row, highlighted in light blue. This proves $q \implies p \equiv \sim p \implies \sim q$.
- Also notice that the "Original" column is different from the "Converse" and "Inverse" columns, confirming their non-equivalence.
Example 1. Consider the statement "If a quadrilateral is a square, then it has four right angles." ($p \implies q$). Write its converse, inverse, and contrapositive. Determine the truth value of all four statements.
Answer:
Let $p$: A quadrilateral is a square.
Let $q$: A quadrilateral has four right angles.
The negations are:
$\sim p$: A quadrilateral is not a square.
$\sim q$: A quadrilateral does not have four right angles.
Original Statement ($p \implies q$): "If a quadrilateral is a square, then it has four right angles."
Truth Value:
A square is defined as having four equal sides and four right angles. So, if a quadrilateral is a square ($p$ is True), it must have four right angles ($q$ is True). T $\implies$ T is True. If a quadrilateral is not a square ($p$ is False), the implication is true regardless of $q$. The statement is True.
Truth Value of Original: T
Converse ($q \implies p$): "If a quadrilateral has four right angles, then it is a square."
Truth Value:
If a quadrilateral has four right angles ($q$ is True), is it necessarily a square ($p$ is True)? No. A rectangle has four right angles but is not necessarily a square (sides might be unequal). For a non-square rectangle (e.g., sides 2x3), $q$ is True, but $p$ is False. T $\implies$ F is False.
Truth Value of Converse: F
Inverse ($\sim p \implies \sim q$): "If a quadrilateral is not a square, then it does not have four right angles."
Truth Value:
If a quadrilateral is not a square ($\sim p$ is True), does it necessarily not have four right angles ($\sim q$ is True)? No. Consider a rectangle that is not a square (sides 2x3). It is not a square ($\sim p$ is True). But it *does* have four right angles ($\sim q$ is False). T $\implies$ F is False.
Truth Value of Inverse: F
Note that the Converse and Inverse have the same truth value (both False), confirming their logical equivalence in this case.
Contrapositive ($\sim q \implies \sim p$): "If a quadrilateral does not have four right angles, then it is not a square."
Truth Value:
If a quadrilateral does not have four right angles ($\sim q$ is True), can it possibly be a square? No, because a square must have four right angles. So, if $\sim q$ is True, $\sim p$ must also be True. T $\implies$ T is True. If $\sim q$ is False, the implication is True. The statement is True.
Truth Value of Contrapositive: T
Note that the Original and Contrapositive have the same truth value (both True), confirming their logical equivalence in this case.
Summary of truth values for this example: Original (T), Converse (F), Inverse (F), Contrapositive (T). This clearly shows the pairings: (Original, Contrapositive) and (Converse, Inverse).
Competitive Exam Pointer: Related Conditionals Summary
This section is high-yield for logical reasoning questions. Master these points:
- Definitions: Be able to form the Converse ($q \implies p$), Inverse ($\sim p \implies \sim q$), and Contrapositive ($\sim q \implies \sim p$) from $p \implies q$.
- Key Equivalences:
- **Original $\equiv$ Contrapositive** ($p \implies q \equiv \sim q \implies \sim p$) - **Most Important!**
- **Converse $\equiv$ Inverse** ($q \implies p \equiv \sim p \implies \sim q$)
- Non-Equivalences: Original is NOT equivalent to Converse, and Original is NOT equivalent to Inverse.
- Truth Tables: Understand how to build and use the truth table for all four statements to verify relationships or evaluate specific cases.
- Proof Technique: Recognize that proving the contrapositive of a statement is a valid way to prove the original statement.
- Application: Be able to take a statement in English, identify $p$ and $q$, translate it into logical form, and then write its converse, inverse, and contrapositive both symbolically and back in English. Determine their truth values based on known facts or provided conditions.
Remember the pairs: Original and Contrapositive are a logically equivalent pair. Converse and Inverse are a logically equivalent pair.
Logical Equivalences: Arrows connect equivalent statements.