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BITSAT 2024Paper 1
Exam
BITSAT
Year
2024
Paper
Paper 1
Subject
Mathematics
Chapter
Mathematical Reasoning

BITSAT 2024 · Paper 1 · Mathematics · Mathematical Reasoning

Consider the following two propositions: P_1: (p arrow q)P_2:(p q)...

Consider the following two propositions: P1:∼(p→∼q)P2:(p∧∼q)∧((∼p)∨q)\mathrm{P}_1: \sim(p \rightarrow \sim q)P_2:(p \wedge \sim q) \wedge((\sim p) \vee q) If the proposition p→((∼p)∨q)p \rightarrow((\sim p) \vee q) is evaluated as FALSE, then :

Options

  1. AP1P_1 is TRUE and P2P_2 is FALSE
  2. BP1\mathrm{P}_1 is FALSE and P2\mathrm{P}_2 is TRUE
  3. CBoth P1P_1 and P2P_2 are FALSE
  4. DBoth P1P_1 and P2P_2 are TRUE

Correct answer

C. Both P1P_1 and P2P_2 are FALSE

Step-by-step solution

Given propositions P1:∼(p→∼q)P2:(p∧∼q)∧((∼p)∨q)\mathrm{P}_1: \sim(\mathrm{p} \rightarrow \sim \mathrm{q})\mathrm{P}_2:(p \wedge \sim q) \wedge((\sim p) \vee q) Required table is shown below. <tableclass="qx−match−listqx−match−table"border="1"cellpadding="6"cellspacing="0"><tr><td>p</td><td>q</td><td>∼p</td><td>∼q</td><td>∼p∨q</td><td>p→(∼(p∨q)</td><td>p→∼q</td><td>∼(p→∼q)</td><td>p∧∼q</td><td>p2</td></tr><tr><td>TTFFTTFTFF</td></tr><tr><td>TFFTFFTFTF</td></tr><tr><td>FTTFTTTFFF</td></tr><tr><td>FFTTTTTFFF</td></tr></table><table class="qx-match-list qx-match-table" border="1" cellpadding="6" cellspacing="0"><tr><td>\mathbf{p}</td><td>\mathbf{q}</td><td>\sim p</td><td>\sim q</td><td>\sim p \vee q</td><td>p \rightarrow(\sim(p \vee q)</td><td>p \rightarrow \sim q</td><td>\sim(p \rightarrow \sim q)</td><td>p \wedge \sim q</td><td>p_2</td></tr><tr><td>T T F F T T F T F F</td></tr><tr><td>T F F T F F T F T F</td></tr><tr><td>F T T F T T T F F F</td></tr><tr><td>F F T T T T T F F F</td></tr></table>) (p \rightarrow(\simp \vee q)) is FF when pp is true qq is false From table (P_1 \& P_2) both are false
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