Verified Solution • 2025–26 Session moderate

QJEEM23Mae7f0: JEE Main 2023 Mathematics - Class 11 JEEM23Mae7f0 Solution: Let p and q be two statements. Then ∼ ( p ∧ ( p ⇒ ∼ q ) ) is…

Let p and q be two statements. Then ∼(p∧(p⇒∼q)) is equivalent to

Multiple Choice Options

  • A (∼p)∨q
  • B p∨(p∧(∼q))
  • C p∨(p∧q)
  • D p∨((∼p)∧q)

Step-by-Step Verified Solution

Making truth table (E≡∼(p∧(p⇒∼q))(E \equiv \sim(p \wedge(p \Rightarrow \sim q))

pp qq ∼p\sim p ∼q\sim q p∨qp \vee q p∧qp \wedge q <mjx−containerclass="MathJaxCtxtMenuAttached0"jax="CHTML"display="true"tabindex="0"ctxtmenucounter="5"style="font−size:129.2<mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" display="true" tabindex="0" ctxtmenu_counter="5" style="font-size: 129.2%; position: relative;"><mjx-math display="true" class="MJX-TEX" aria-hidden="true" style="margin-left: 0px; margin-right: 0px;"><mjx-mtable style="min-width: 4.355em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align: right;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c21D2"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c223C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D45E TEX-I"></mjx-c></mjx-mi><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" display="true" tabindex="0" ctxtmenu_counter="25" style="font-size: 129.1%; position: relative;"><mjx-math display="true" class="MJX-TEX" aria-hidden="true" style="margin-left: 0px; margin-right: 0px;"><mjx-mtable style="min-width: 4.355em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align: right;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c21D2"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c223C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D45E TEX-I"></mjx-c></mjx-mi><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true" columnalign="right" columnspacing="" rowspacing="3pt"><mtr><mtd><mi>p</mi><mo stretchy="false">⇒</mo><mi>p</mi><mo>∼</mo><mi>q</mi></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container> p∧(p⇒∼q)p \wedge (p \Rightarrow \sim q) E\mathrm{E}
T\mathrm{T} T\mathrm{T} F\mathrm{F} F\mathrm{F} T\mathrm{T} T\mathrm{T} F\mathrm{F} F\mathrm{F} T\mathrm{T}
T\mathrm{T} F\mathrm{F} F\mathrm{F} T\mathrm{T} T\mathrm{T} F\mathrm{F} T\mathrm{T} T\mathrm{T} F\mathrm{F}
F\mathrm{F} T\mathrm{T} T\mathrm{T} F\mathrm{F} T\mathrm{T} F\mathrm{F} T\mathrm{T} F\mathrm{F} T\mathrm{T}
F\mathrm{F} F\mathrm{F} T\mathrm{T} T\mathrm{T} F\mathrm{F} F\mathrm{F} T\mathrm{T} F\mathrm{F} T\mathrm{T}

&amp;\&amp;

∼p∨q\sim p \vee q p∨(∼p∧q)p \vee(\sim p \wedge q) p∨(p∧q)p \vee(p \wedge q) p∨(p∧∼q)p \vee(p \wedge \sim q)
T\mathrm{T} T\mathrm{T} T\mathrm{T} T\mathrm{T}
F\mathrm{F} T\mathrm{T} T\mathrm{T} T\mathrm{T}
T\mathrm{T} T\mathrm{T} F\mathrm{F} F\mathrm{F}
T\mathrm{T} F\mathrm{F} F\mathrm{F} F\mathrm{F}

∴∼(p∧(p⇒∼q))\therefore \sim(p \wedge(p \Rightarrow \sim q)) is equivalent to ∼p∨q\sim p \vee q

Other Method :

∼(p∨(p→∼q)

=∼p∨∼(p→∼q)

=∼p∨∼(∼p∨∼q)

=∼p∨(p∧q)

=(∼p∨p)∧(∼p∨q)

=t∧(∼p∨q)

=∼p∨q

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