Tag: implications

Questions Related to implications

Identify the Law of Logic
$p \wedge T \equiv T$
$p \vee F \equiv F$

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

  1. Complement Law

  2. Identity Law

  3. Involution Law

  4. Absorption Law

Show answer
Correct Option: B
Explanation:

Identity Law

Identity law observes how certain expression will behave when one of the terms is fixed
$p \vee F \equiv F$ and $p\wedge T\equiv T$

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

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

  1. True

  2. False

Show answer
Correct Option: A
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

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

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

  1. Associative law

  2. Commutative Law

  3. Involution Law

  4. Conditional Law

Show answer
Correct Option: A
Explanation:

Associative Law

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

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

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

  1. Commutative Law

  2. DeMorgan's Law

  3. Complement Law

  4. Conditional Law

Show answer
Correct Option: B
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

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

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

  1. $\sim p \vee q$

  2. $ p \wedge \sim q$

  3. $\sim p \wedge q$

  4. $ p \vee q$

Show answer
Correct Option: C
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

$p\rightarrow q$ is equivalent to

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

  1. $\sim p\vee \sim q$

  2. $ p\vee \sim q$

  3. $\sim p\vee q$

  4. $\sim p\wedge q$

Show answer
Correct Option: C
Explanation:
$p$ $q$ $p\rightarrow q$ $\sim p\vee q$
$T$ $T$ $T$ $T$
$T$ $F$ $T$ $T$
$F$ $F$ $T$ $T$
$F$ $T$ $T$ $T$

$(p\rightarrow q)\longleftrightarrow (\sim p\vee q)$ is a tautology.

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

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

  1. $p \leftrightarrow q$

  2. $p \rightarrow q$

  3. $p \leftrightarrow \sim q$

  4. $\sim p \rightarrow q$

Show answer
Correct Option: A
Explanation:
given statement 
$(p\wedge q)\vee (\sim p\wedge \sim q)$
$(p\wedge q)\vee (\sim p\wedge \sim q)$
$(p\wedge q)\vee (\sim(p\wedge  q))$
$(p\wedge q)\wedge(p\wedge  q)$
$p\leftrightarrow q$

$p \leftrightarrow q \equiv ?$

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

  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

Show answer
Correct Option: D
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) $

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

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

  1. Conditional Law

  2. Demorgan's Law

  3. Absorption Law

  4. Identity Law

Show answer
Correct Option: B
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

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

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

  1. Idempotent Law

  2. Conditional Law

  3. Involution Law

  4. Commutative Law

Show answer
Correct Option: B
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.