Tag: implications

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Multiple choice maths proofs in mathematics implications principle of mathematical induction different forms of theoretical statements

Statement I : if p is false statement and q is true statement, then $ \sim \,p\, \wedge \,q$ is true
Statement II : $ \sim \,p\, \wedge \,q$ is equivalent to $ \sim \left( {pV \sim \,q\,} \right)$

  1. Statement I is true and Statement II is the correct explanation for statement.

  2. Statement I is true and Statement II is true. Statement II is not the correct explanation for the Statement I.

  3. Statement I is true but Statement II is false.

  4. Statement I is false but Statement II is true
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
p q $ \sim p$ $ \sim q$ (i)$ \sim \,p\, \wedge \,q$ $p\,\, \vee \, \sim \,q$ (ii)$ \sim \left( {p\,\, \vee \, \sim \,q} \right)$
T T F F F T F
T F F T F T F
F T T F T F T
F F T T F T F

p is false
q is true
$ \sim p\, \wedge \,q$ is true
from (i) & (ii)
$ \sim p\, \wedge \,q$ = $ \sim \left( {p\,\, \vee  \sim \,\,q} \right)$
Statement (i) & (ii) is correct 
$ \sim \left( {p\,\, \vee  \sim \,\,q} \right)\, = \, \sim \,p\, \wedge \, \sim \,\left( { \sim \,q} \right)$
$ = \, \sim \,p\, \wedge \,\,q$

Multiple choice maths proofs in mathematics implications principle of mathematical induction different forms of theoretical statements

$ \sim (p \vee q) \vee ( \sim p \wedge q)$ is logically euivalent to 

  1. $ \sim p$

  2. p

  3. q
  4. $ \sim q$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

${ \sim  }({ p }{ \vee  }{ q }){ \vee  }({ \sim  }{ p }{ \wedge  }{ q })\ ={ \sim  }({ U }){ \vee  }({ q }{ \wedge  }{ q })\ ={ \phi  }{ \vee  }({ q })\ ={ q }$


Where U is universal set , $\phi $ is nulll set and p and q are two disjoint sets 
Correct option is C

Multiple choice maths proofs in mathematics implications principle of mathematical induction different forms of theoretical statements

Cost of a diamond varies directly as the square of its weight.A diamond broke into four pieces with their weight in the ratio $1:2:3:4$ If the loss in the total value of the diamond was $Rs.\ 70000$. Find the prices of the original diamond.

  1. $10000\ Rs$
  2. $100000\ Rs$
  3. $1000000\ Rs$
  4. $2000000\ Rs$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let, the weights of the four pieces of the diamond $x,2x,3x,4x$ respectively.

$\therefore$ Total weight of the original diamond$=x+2x+3x+4x=10x$
Let, price of the original diamond$=k{ (100{ x }^{ 2 }) },where\quad k\quad is\quad constant$
$\therefore$ Cost of four pieces$=k({ x }^{ 2 }+4{ x }^{ 2 }+9{ x }^{ 2 }+16{ x }^{ 2 })\ =k(30{ x }^{ 2 })$
Loss in the total value of the diamond,
$k(100{ x }^{ 2 })-k(30{ x }^{ 2 })=70000\ \Rightarrow k(70{ x }^{ 2 })=70000\ \Rightarrow k{ x }^{ 2 }=Rs.1000$
Hence, price of the original diamond$=k(100{ x }^{ 2 })\ =1000\times 100=Rs.100000.$

Multiple choice maths proofs in mathematics implications principle of mathematical induction different forms of theoretical statements

Which of the following is always true ? 

  1. $\left( {p \to q} \right) \cong \left( { \sim q \to \sim p} \right)$
  2. $ \sim \left( {p \vee q} \right) \cong \left( { \sim p \vee \sim q} \right)$
  3. $ \sim \left( {p \to q} \right) \cong \left( {p \vee \sim q} \right)$
  4. $ \sim \left( {p \wedge q} \right) \cong \left( { \sim p \wedge \sim q} \right)$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

We know that 


$p\rightarrow q\equiv \sim p\wedge q$

$\sim (p\rightarrow q)\equiv \sim (\sim p\wedge q)$

$\sim (p\rightarrow q)\equiv p\vee \sim q$               (De morgan's law)

$C$ is correct

Multiple choice maths proofs in mathematics implications principle of mathematical induction different forms of theoretical statements

$p \leftrightarrow q \equiv  \sim \left( {p\Delta  \sim q} \right)\Delta  \sim \left( {q\Delta  \sim p} \right)$

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The expression uses the delta symbol, which usually denotes XOR. The identity p <-> q is equivalent to the negation of XOR, and the expression provided simplifies to this identity.

Multiple choice maths proofs in mathematics implications principle of mathematical induction different forms of theoretical statements

Let $p$ and $q$ be two statements, then $ \sim ( \sim p \wedge q) \wedge (p \vee q)$ is logically equivalent to 

  1. $q$
  2. $p\vee q$
  3. $p$
  4. $p\vee \sim q$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Using De Morgan's laws, not(not p and q) becomes p or not q. Distributing this with (p or q) yields p or (not q and q), which simplifies to p or False, resulting in p.

Multiple choice maths proofs in mathematics implications principle of mathematical induction different forms of theoretical statements

$ \sim (p \wedge q) \to ( \sim p \vee ( \sim p \vee q))$  is equivalent to 

  1. $p \vee \sim q$
  2. $p \wedge \sim q$
  3. $ \sim p \vee q$
  4. $ \sim p \wedge q$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

An implication A to B is equivalent to not A or B. Here, A is not(p and q), so not A is (p and q). B is not p or q. Thus, the expression becomes (p and q) or not p or q, which simplifies using absorption laws to not p or q.