Tag: different forms of theoretical statements

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Questions Related to different forms of theoretical statements

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

Is $(p\rightarrow q)\vee (q\rightarrow p)$  a tautology ?

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
$p$ $q$ $(p\rightarrow q)$ $(q\rightarrow p)$ $(p\rightarrow q)\vee(q\rightarrow p)$
T T             T              T                               T
T F             F              T                               T
F T             T              F                               T
F F             T              T                               T               

The given statement is a tautology as the truth table has all the values as true in the output which is the property of tautology
Multiple choice maths proofs in mathematics implications principle of mathematical induction different forms of theoretical statements

Identify the Law of Logic
$(p \vee q) \vee r \equiv p \vee (q \vee r) \equiv p \vee q \vee r$

  1. Associative law

  2. Commutative Law

  3. Involution Law

  4. Conditional Law
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Associative Law

This law allows the removal of brackets from an expression and regrouping of the variables.
$(p\vee q)\vee r \equiv p \vee (q \vee r)\equiv p\vee q\vee r$

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

Identify the Law of Logic
$\sim(p \wedge q) \equiv \sim p \vee \sim q$

  1. Commutative Law

  2. DeMorgan's Law

  3. Complement Law

  4. Conditional Law
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
Given 
$\sim (p\wedge q)=\sim p \vee \sim q$

It is Demorgan's law 
according to the if we take transpose or negation of any quatity then all the relation get opposite

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

The equivalent statement of $(p \vee q) \wedge \sim p$ is?

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

$(p\vee q)\wedge \sim p$
$=(p\wedge \sim p)\vee (q\wedge \sim p)$ Distributive Law
$=F\vee (q\wedge \sim p)$ Negation Law
$=(q\wedge \sim p)$ Identity Law
$=(\sim p \wedge q)$ Commutative Law

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

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

  1. $p \leftrightarrow q$
  2. $p \rightarrow q$
  3. $p \leftrightarrow \sim q$
  4. $\sim p \rightarrow q$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
given statement 
$(p\wedge q)\vee (\sim p\wedge \sim q)$
$(p\wedge q)\vee (\sim p\wedge \sim q)$
$(p\wedge q)\vee (\sim(p\wedge  q))$
$(p\wedge q)\wedge(p\wedge  q)$
$p\leftrightarrow q$

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

$p \leftrightarrow q \equiv ?$

  1. $\sim (p \vee \sim q) \wedge \sim(p \wedge \sim q)$
  2. $\sim (p \wedge \sim q) \wedge \sim(p \wedge \sim q)$
  3. $\sim (p \wedge \sim q) \wedge \sim(p \vee \sim q)$

  4. None of these
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$p\leftrightarrow q=(p\rightarrow q)\wedge (q \rightarrow p)$
We know that $(p\rightarrow q)=(\sim p \vee q)$
So,
$(p\rightarrow q)\wedge (q \rightarrow p)=(\sim p \vee q)\wedge (\sim q \vee p)$
Now, apply the De'morgan law state that $\sim(p\vee q)= (\sim p \wedge \sim q)$
Therefore,
$(\sim p \vee q)\wedge (\sim q \vee p)=\sim (p \wedge \sim q) \wedge \sim (q\wedge \sim p) $

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

Identify the Law of Logic
$\sim(p \vee q) \equiv \sim p \wedge \sim q$

  1. Conditional Law

  2. Demorgan's Law

  3. Absorption Law

  4. Identity Law
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
Given 
$\sim (p\wedge q)=\sim p \vee \sim q$

It is Demorgan's law 
according to the if we take transpose or negation of any quatity then all the relation get opposite

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

Identify the Law of Logic
$p \rightarrow q \equiv \sim p \vee q$

  1. Idempotent Law

  2. Conditional Law

  3. Involution Law

  4. Commutative Law
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
|  $p$ |  $q$ |  $p\rightarrow q$ |  $\sim p$ |  $(\sim p)\vee q$ | | --- | --- | --- | --- | --- | |  $T$ |   $T$ |   $T$ |   $F$ |   $T$ | |   $T$ |   $F$ |   $F$ |   $F$ |   $F$ | |  $F$ |   $T$ |   $T$ |   $T$ |   $T$ | |   $F$ |   $F$ |   $T$ |   $T$ |   $T$ |
We can say that if $p$ ,then $q$ or $p$ implies $q$ .
'$\rightarrow$' is called a conditional operator.
So, the giving logical equivalence is the conditional law.