Tag: principle of mathematical induction

Question 1

Let p and q be any two logical statements and $r : p \rightarrow (\sim p \vee q)$. If r has a truth value F, then the truth values of p and q are respectively

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

F, F
T, T
F, T
T, F

Explanation:

p	q	$\sim$p	$\sim$ p $\vee$ q	r
T	T	F	T	T
F	F	T	T	T
T	F	F	F	F
F	T	T	T	T

$\therefore$ Clearly from above able, If r has a truth value F, then the truth values of p and core T and F respectively.

Answer

Correct Option: D

Question 2

State, whether the is given the statement, is True or False.
$\sim [(p \vee \sim q) \rightarrow (p \wedge \sim q)] \equiv (p \vee \sim q) \wedge (\sim \vee q)$

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

True
False

Explanation:

$p$	$q$	$\sim q$	$(p\vee \sim q)$	$(p\wedge \sim q)$	$(p\vee \sim q)\rightarrow (p\wedge \sim q)$	$\sim[(p\vee \sim q)\rightarrow (p\wedge \sim q)]$
$T$	$T$	$F$	$T$	$F$	$F$	$T$
$T$	$F$	$T$	$T$	$T$	$T$	$F$
$F$	$T$	$F$	$F$	$F$	$T$	$F$
$F$	$F$	$T$	$T$	$F$	$F$	$T$

$p$	$q$	$\sim q$	$(p\vee \sim q)$	$\sim p$	$(\sim p \vee q)$	$(p\vee \sim q)\wedge (\sim p \vee q)$
$T$	$T$	$F$	$T$	$F$	$T$	$T$
$T$	$F$	$T$	$T$	$F$	$F$	$F$
$F$	$T$	$F$	$F$	$T$	$T$	$F$
$F$	$F$	$T$	$T$	$T$	$T$	$T$

Answer

Correct Option: A

Question 3

$\sim (p \vee q) \vee (\sim p \wedge q) \equiv ?$

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

$\sim q$
$q$
$\sim p$
$p$

Answer

Correct Option: C

Question 4

State whether the following statements is True or False?
$p \leftrightarrow q \equiv (p \wedge q) \vee (\sim p \wedge \sim q)$

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

True
False

Answer

Correct Option: A

Question 5

Let p,q be statements. Negation of statement $p \leftrightarrow ~ q$, is

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

$~ q \rightarrow p$
$ ~ p v q$
$p \leftrightarrow q$
$p \rightarrow q$

Explanation:

p	q	~q	$p \leftrightarrow ~q$	$~(p \leftrightarrow ~q)$	$~q \rightarrow p$	$~p v q$	$p \leftrightarrow q$	$p \rightarrow q$
T	T	F	F	T	T	T	T	T
T	F	T	T	F	T	F	F	F
F	T	T	T	F	T	T	F	T
F	F	F	F	T	F	T	T	T

Answer

Correct Option: A

$p$	$q$	$\sim q$	$(p\vee \sim q)$	$(p\wedge \sim q)$	$(p\vee \sim q)\rightarrow (p\wedge \sim q)$	$\sim[(p\vee \sim q)\rightarrow (p\wedge \sim q)]$
$T$	$T$	$F$	$T$	$F$	$F$	$T$
$T$	$F$	$T$	$T$	$T$	$T$	$F$
$F$	$T$	$F$	$F$	$F$	$T$	$F$
$F$	$F$	$T$	$T$	$F$	$F$	$T$

$p$	$q$	$\sim q$	$(p\vee \sim q)$	$\sim p$	$(\sim p \vee q)$	$(p\vee \sim q)\wedge (\sim p \vee q)$
$T$	$T$	$F$	$T$	$F$	$T$	$T$
$T$	$F$	$T$	$T$	$F$	$F$	$F$
$F$	$T$	$F$	$F$	$T$	$T$	$F$
$F$	$F$	$T$	$T$	$T$	$T$	$T$

$p$	$q$	$\sim q$	$(p\vee \sim q)$	$(p\wedge \sim q)$	$(p\vee \sim q)\rightarrow (p\wedge \sim q)$	$\sim[(p\vee \sim q)\rightarrow (p\wedge \sim q)]$
$T$	$T$	$F$	$T$	$F$	$F$	$T$
$T$	$F$	$T$	$T$	$T$	$T$	$F$
$F$	$T$	$F$	$F$	$F$	$T$	$F$
$F$	$F$	$T$	$T$	$F$	$F$	$T$

$p$	$q$	$\sim q$	$(p\vee \sim q)$	$\sim p$	$(\sim p \vee q)$	$(p\vee \sim q)\wedge (\sim p \vee q)$
$T$	$T$	$F$	$T$	$F$	$T$	$T$
$T$	$F$	$T$	$T$	$F$	$F$	$F$
$F$	$T$	$F$	$F$	$T$	$T$	$F$
$F$	$F$	$T$	$T$	$T$	$T$	$T$

Tag: principle of mathematical induction

Questions Related to principle of mathematical induction

$p$	$q$	$\sim q$	$(p\vee \sim q)$	$(p\wedge \sim q)$	$(p\vee \sim q)\rightarrow (p\wedge \sim q)$	$\sim[(p\vee \sim q)\rightarrow (p\wedge \sim q)]$
$T$	$T$	$F$	$T$	$F$	$F$	$T$
$T$	$F$	$T$	$T$	$T$	$T$	$F$
$F$	$T$	$F$	$F$	$F$	$T$	$F$
$F$	$F$	$T$	$T$	$F$	$F$	$T$

$p$	$q$	$\sim q$	$(p\vee \sim q)$	$\sim p$	$(\sim p \vee q)$	$(p\vee \sim q)\wedge (\sim p \vee q)$
$T$	$T$	$F$	$T$	$F$	$T$	$T$
$T$	$F$	$T$	$T$	$F$	$F$	$F$
$F$	$T$	$F$	$F$	$T$	$T$	$F$
$F$	$F$	$T$	$T$	$T$	$T$	$T$