6.4 Add Subtract Multiply Polynomials Practice .pdf - IM1 - Task ... - Free Printable

Problem Analysis:

The task involves analyzing the given logical statements and determining their truth values based on the provided conditions. The problem is structured as a multiple-choice question where we need to evaluate which of the given options is correct.

#### Given Information:
1. Logical Statements:
- $ p \rightarrow q $
- $ r \rightarrow s $
- $ t \rightarrow u $

2. Negations:
- $ \neg p \rightarrow \neg q $
- $ \neg r \rightarrow \neg s $
- $ \neg t \rightarrow \neg u $

3. Conjunctions:
- $ (p \wedge r) \rightarrow (q \wedge s) $
- $ (r \wedge t) \rightarrow (s \wedge u) $
- $ (t \wedge p) \rightarrow (u \wedge q) $

4. Disjunctions:
- $ (p \vee r) \rightarrow (q \vee s) $
- $ (r \vee t) \rightarrow (s \vee u) $
- $ (t \vee p) \rightarrow (u \vee q) $

5. Implications with Negations:
- $ (\neg p \wedge \neg r) \rightarrow (\neg q \wedge \neg s) $
- $ (\neg r \wedge \neg t) \rightarrow (\neg s \wedge \neg u) $
- $ (\neg t \wedge \neg p) \rightarrow (\neg u \wedge \neg q) $

6. Disjunctions with Negations:
- $ (\neg p \vee \neg r) \rightarrow (\neg q \vee \neg s) $
- $ (\neg r \vee \neg t) \rightarrow (\neg s \vee \neg u) $
- $ (\neg t \vee \neg p) \rightarrow (\neg u \vee \neg q) $

#### Task:
We need to determine which of the following options is true:
1. $ (p \wedge r) \rightarrow (q \wedge s) $
2. $ (r \wedge t) \rightarrow (s \wedge u) $
3. $ (t \wedge p) \rightarrow (u \wedge q) $
4. $ (\neg p \wedge \neg r) \rightarrow (\neg q \wedge \neg s) $
5. $ (\neg r \wedge \neg t) \rightarrow (\neg s \wedge \neg u) $
6. $ (\neg t \wedge \neg p) \rightarrow (\neg u \wedge \neg q) $

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Solution Approach:

To solve this, we will analyze each statement systematically using the rules of logic:

#### 1. Implication Rules:
- $ A \rightarrow B $ is false only when $ A $ is true and $ B $ is false.
- For conjunctions ($ \wedge $), both components must be true for the whole statement to be true.
- For disjunctions ($ \vee $), at least one component must be true for the whole statement to be true.

#### 2. Negation Rules:
- $ \neg A $ means "not $ A $."
- If $ A \rightarrow B $ is true, then $ \neg B \rightarrow \neg A $ is also true (contrapositive).

#### 3. Analyzing Each Option:

##### Option 1: $ (p \wedge r) \rightarrow (q \wedge s) $
- This statement says: "If $ p $ and $ r $ are both true, then $ q $ and $ s $ are both true."
- This is logically valid because if $ p \rightarrow q $ and $ r \rightarrow s $ are true, then whenever $ p $ and $ r $ are true, $ q $ and $ s $ must also be true.
- This option is true.

##### Option 2: $ (r \wedge t) \rightarrow (s \wedge u) $
- This statement says: "If $ r $ and $ t $ are both true, then $ s $ and $ u $ are both true."
- This is logically valid because if $ r \rightarrow s $ and $ t \rightarrow u $ are true, then whenever $ r $ and $ t $ are true, $ s $ and $ u $ must also be true.
- This option is true.

##### Option 3: $ (t \wedge p) \rightarrow (u \wedge q) $
- This statement says: "If $ t $ and $ p $ are both true, then $ u $ and $ q $ are both true."
- This is logically valid because if $ t \rightarrow u $ and $ p \rightarrow q $ are true, then whenever $ t $ and $ p $ are true, $ u $ and $ q $ must also be true.
- This option is true.

##### Option 4: $ (\neg p \wedge \neg r) \rightarrow (\neg q \wedge \neg s) $
- This statement says: "If $ p $ and $ r $ are both false, then $ q $ and $ s $ are both false."
- This is not necessarily true. Even if $ p $ and $ r $ are false, $ q $ and $ s $ could still be true (since $ p \rightarrow q $ and $ r \rightarrow s $ do not guarantee that $ q $ and $ s $ are false when $ p $ and $ r $ are false).
- This option is false.

##### Option 5: $ (\neg r \wedge \neg t) \rightarrow (\neg s \wedge \neg u) $
- This statement says: "If $ r $ and $ t $ are both false, then $ s $ and $ u $ are both false."
- This is not necessarily true. Even if $ r $ and $ t $ are false, $ s $ and $ u $ could still be true (since $ r \rightarrow s $ and $ t \rightarrow u $ do not guarantee that $ s $ and $ u $ are false when $ r $ and $ t $ are false).
- This option is false.

##### Option 6: $ (\neg t \wedge \neg p) \rightarrow (\neg u \wedge \neg q) $
- This statement says: "If $ t $ and $ p $ are both false, then $ u $ and $ q $ are both false."
- This is not necessarily true. Even if $ t $ and $ p $ are false, $ u $ and $ q $ could still be true (since $ t \rightarrow u $ and $ p \rightarrow q $ do not guarantee that $ u $ and $ q $ are false when $ t $ and $ p $ are false).
- This option is false.

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Final Answer:

The correct options are:
1. $ (p \wedge r) \rightarrow (q \wedge s) $
2. $ (r \wedge t) \rightarrow (s \wedge u) $
3. $ (t \wedge p) \rightarrow (u \wedge q) $

Thus, the final answer is:
\[
\boxed{1, 2, 3}
\]