You have given following statements. p: 9 × 5 = 45, q: Pune is in Maharashtra, r: 3 is smallest prime number. Then write truth values by activity. i) (p ∧ q) ∧ r = ____________ ii)... You have given following statements. p: 9 × 5 = 45, q: Pune is in Maharashtra, r: 3 is smallest prime number. Then write truth values by activity. i) (p ∧ q) ∧ r = ____________ ii) ~[p ∧ r] = ____________ iii) p → q = ____________ iv) p → r = ____________ Represent following statement by Venn diagram.
Understand the Problem
The question is asking us to evaluate logical statements involving truth values of propositions that are either true or false. We need to complete truth tables based on given propositions about multiplication and geographical statements.
Answer
1. a) $T$ b) $T$ 2. i) $F$ 3. ii) $T$ 4. iii) $T$ 5. iv) $F$
Answer for screen readers
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a) $p \lor q = T$
b) $p \land q = T$
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i) $(p \land q) \land r = F$
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ii) $~[p \land r] = T$
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iii) $p \rightarrow q = T$
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iv) $p \rightarrow r = F$
Steps to Solve
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Define the Propositions and Their Truth Values Here, we define the propositions given in the question.
- $p$: "9 × 5 = 45" (True, T)
- $q$: "Pune is in Maharashtra" (True, T)
- $r$: "3 is the smallest prime number" (False, F)
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Evaluate Logical Connectives for p ∨ q (p OR q)
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If both $p$ and $q$ are true:
- $p \lor q = T \lor T = T$
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If both $p$ and $q$ are false:
- $p \lor q = F \lor F = F$
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If both $p$ and $q$ are true:
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Evaluate Logical Connectives for p ∧ q (p AND q)
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If both $p$ and $q$ are true:
- $p \land q = T \land T = T$
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If both $p$ and $q$ are false:
- $p \land q = F \land F = F$
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If both $p$ and $q$ are true:
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Fill Out the Truth Table for (p ∧ q) ∧ r
- With $p = T$, $q = T$, and $r = F$:
- $(p \land q) \land r = (T \land T) \land F = T \land F = F$
- With $p = T$, $q = T$, and $r = F$:
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Evaluate Negation: ~[p ∧ r]
- The statement $p \land r = T \land F = F$:
- So, $~(p \land r) = ~F = T$
- The statement $p \land r = T \land F = F$:
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Evaluate Implication Statements: p → q and p → r
- For $p \rightarrow q$:
- If $p$ is true, then $q$ is also true
- Thus, $p \rightarrow q = T \rightarrow T = T$
- For $p \rightarrow r$:
- Here, $p$ is true but $r$ is false
- Thus, $p \rightarrow r = T \rightarrow F = F$
- For $p \rightarrow q$:
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Evaluate Biconditional Statements: p ↔ q
- Here, we check if both are true:
- Since both $p$ and $q$ are true, $p \leftrightarrow q = T \leftrightarrow T = T$
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a) $p \lor q = T$
b) $p \land q = T$
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i) $(p \land q) \land r = F$
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ii) $~[p \land r] = T$
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iii) $p \rightarrow q = T$
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iv) $p \rightarrow r = F$
More Information
This evaluation involves using logical operations like AND ($\land$), OR ($\lor$), NOT ($\sim$), and implications ($\rightarrow$) to determine the truth values based on the initial propositions. The truth table helps visualize how different combinations of truth values affect the outcomes.
Tips
- Not understanding the distinction between AND and OR operations, leading to incorrect conclusions.
- Misinterpreting the logic in implications ($\rightarrow$) and biconditionals ($\leftrightarrow$); remember that implication is false only when the first statement is true and the second false.
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