Practice worksheet for identifying the converse, inverse, and contrapositive of conditional statements.

Let's solve each conditional statement by finding its converse, inverse, and contrapositive.

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1. If it is snowing, then the temperature is 32°F or less.

This is a conditional statement in the form:
If P, then Q
Where:
- P = It is snowing
- Q = The temperature is 32°F or less

#### Converse:
Switch the hypothesis and conclusion.
If the temperature is 32°F or less, then it is snowing.
(If Q, then P)

> Note: This is not necessarily true — cold temperatures don’t always mean it’s snowing.

#### Inverse:
Negate both the hypothesis and conclusion.
If it is not snowing, then the temperature is greater than 32°F.
(If not P, then not Q)

> Again, this isn't necessarily true — it might not be snowing even if it's below freezing (e.g., raining).

#### Contrapositive:
Negate and switch the hypothesis and conclusion.
If the temperature is greater than 32°F, then it is not snowing.
(If not Q, then not P)

> This is logically equivalent to the original statement.

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2. If a bug is a insect, then it has 6 legs.

Note: There's a small error in wording — "bug" is often used informally for insects, but technically, all bugs are insects, but not all insects are bugs. However, we'll treat this as:
If an animal is a bug (insect), then it has 6 legs.

So:
P = A bug is an insect
Q = It has 6 legs

But actually, this should be interpreted as:
If an animal is an insect, then it has 6 legs.
(That’s more accurate.)

So let's rephrase:
If an animal is an insect, then it has 6 legs.

Now:

#### Converse:
If an animal has 6 legs, then it is an insect.
(If Q, then P)

> Not necessarily true — some non-insects have 6 legs (e.g., some arachnids? No, wait — arachnids have 8). Actually, many animals with 6 legs are insects, but not all — e.g., some crustaceans or hexapods. But generally, this is close. Still, not logically equivalent.

#### Inverse:
If an animal is not an insect, then it does not have 6 legs.
(If not P, then not Q)

> False — some non-insects (like certain spiders or other arthropods) may have 6 legs? Actually, no — spiders have 8. So maybe this is mostly true, but still not guaranteed.

#### Contrapositive:
If an animal does not have 6 legs, then it is not an insect.
(If not Q, then not P)

> This is logically equivalent to the original statement.
> True: All insects have 6 legs, so if something doesn't have 6 legs, it can't be an insect.

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3. If a number ends in 5, then it is divisible by 5.

P = A number ends in 5
Q = It is divisible by 5

#### Converse:
If a number is divisible by 5, then it ends in 5.
(If Q, then P)

> False! Numbers divisible by 5 end in 0 or 5, not just 5.
> Example: 10 is divisible by 5 but ends in 0.

#### Inverse:
If a number does not end in 5, then it is not divisible by 5.
(If not P, then not Q)

> Also false — numbers ending in 0 (like 10, 20) are divisible by 5 but do not end in 5.

#### Contrapositive:
If a number is not divisible by 5, then it does not end in 5.
(If not Q, then not P)

> True! And logically equivalent to the original.
> If a number isn’t divisible by 5, it can’t end in 5 (since only numbers ending in 0 or 5 are divisible by 5).

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✔ Final Answers:

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1. If it is snowing, then the temperature is 32°F or less.

- Converse: If the temperature is 32°F or less, then it is snowing.
- Inverse: If it is not snowing, then the temperature is greater than 32°F.
- Contrapositive: If the temperature is greater than 32°F, then it is not snowing.

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2. If a bug is an insect, then it has 6 legs.
*(Assuming: If an animal is an insect, then it has 6 legs)*

- Converse: If an animal has 6 legs, then it is an insect.
- Inverse: If an animal is not an insect, then it does not have 6 legs.
- Contrapositive: If an animal does not have 6 legs, then it is not an insect.

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3. If a number ends in 5, then it is divisible by 5.

- Converse: If a number is divisible by 5, then it ends in 5.
- Inverse: If a number does not end in 5, then it is not divisible by 5.
- Contrapositive: If a number is not divisible by 5, then it does not end in 5.

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🔍 Key Notes:

- Only the contrapositive is logically equivalent to the original statement.
- The converse and inverse are not logically equivalent to the original — they may be true or false independently.

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