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1.

If `(p^^~r) to (~p vvq)` is false, then truth values of p,q and r are respectively.A. T,T,TB. T,F,TC. T,F,FD. F,T,T

Answer» Correct Answer - C
`(p^^~r) to (~p vvq)` is false.
thus, `(p^^~r)` is true and `(~pvvq)` is false.
So, (p is true and `~r` is true) and (`~p` is false and q is false)
Therefore, p is true, r is false and q is false
Discussion
2.

Statements `(p to q) harr (~q to ~p)`A. is contradictionB. is tautologyC. is neither contradiction not tautologyD. None of these

Answer» Correct Answer - B
`(~q to ~p)` is contrapositive of `(p to q)`
Therefore, `(pto q) harr(~q to ~p)` is tautology.
Discussion
3.

The contrapositive of inverse of `p to ~ q` isA. `ptoq`B. `~ q to p`C. `q to p`D. `~q to ~p`

Answer» Correct Answer - B
Inverse is `~p to q`
contraspositive is `~p to q` is `~q to p`
Discussion
4.

The contrapositive of `(pvvq) to r` isA. `r to (pvvq)`B. `~r to (pvvq)`C. `~r to (~ p ^^~q)`D. `p to (qvvr)`

Answer» Correct Answer - C
`p to q` is false only when p is true and q is false
Therefore, `p to q` is `~ q to ~p`
Therefore, contrapositive of `(pvvq) to r` is
`~r to ~(pvvq)`
`-=~ rto (~p^^~q)`
Discussion
5.

If the inverse of implication ` p to q ` is defined as ` ~ p to ~q` , then the inverse of the proposition ` ( p ^^ ~ q) to r ` isA. `~r to (~pvvq)`B. `r to (p^^~q)`C. `~qvv(p^^r)`D. none of these

Answer» Correct Answer - C
Inverse of `p to q` is `~p to ~q`
Therefore, inverse of `(p^^~q) to r` is
`~(p^^~q) to ~r`
`-=~(p^^(~(~(p^^r))`
`-=~q^^(p^^r)`
Discussion
6.

Which of the following is logically equivalent to `~(~pto q)`?A. `p^^q`B. `p^^~q`C. `~p^^q`D. `~p^^~q`

Answer» Correct Answer - D
We know that `ptoq -=~pvvq`
`:. ~ p to q-=pvvq`
`:. ~(pto q) -=~ (pvvq)`
`-=~p^^~q`
Discussion
7.

Which of the following is not a proposition ?A. `sqrt(3)` is a primeB. `sqrt(2)` is irrationalC. Mathematics is interestingD. 5 is an even integer

Answer» Correct Answer - C
" Mathematics is interesting" is not a logical sentence. It may be interesting for some persons and may not be interesting for others.
Therefore, this is not a proposition.
Discussion
8.

For the statement: "If a quadilateral is a rectangle , then it has two paisrs of parallel sides", write the converse, inverse and contrapositive statements.

Answer» Converse: If a quadilateral has two pairs of parallel sides, then it is a rectangles
Inverse: IF a quadilateral is not a rectangle, then it does not have two pairs of parallel sides.
Contrapositive: If a quadilateral does not have two pairs of parallel sides, then it is not a rectangle
Here, both converse and inverse are false. since the given statement is true, its contrapositive is true.
Discussion
9.

Prove that `~(~pto ~q) -=~p ^^q`

Answer» `~(~p to ~q) -=~(~(~p)vv~q)`
`-=~(pvv~q)`
`-=~ p^^q`
Discussion
10.

prove that `(p^^q) ^^~(pvvq)` is a contradiction.

Answer» `(p^^q)^^~(pvvq) -=(p^^q)^^(~p^^~q)`
`-=(p^^~p) ^^(q^^~q)`
`-=f^^f`
`-=f`
thus, `(p^^q)^^~(pvvq)` is fallacy, i.e., contradiction.
Discussion
11.

Prove that `~((~p)^^q) -=pvv(~q)`.

Answer» `~((~p) ^^q-=~ (~p)vv~q-=pvv(~q)`
Discussion
12.

`~(pvv(~pvvq))` is equal toA. `~p^^(p^^~q)`. B. `(pvv~q)v~p`C. none of theseD.

Answer» Correct Answer - A
`~[pvv(~pvvq)]`
`-=~p^^~(~pvvq)`
`-=~p^^(~(~p)^^~q)`
`-=~p^^(p^^~p)`
Discussion
13.

The following statement `(p to q) to [(~p to q) to q]` isA. a fallacyB. a tautologyC. equivalent to `~p toq `D. equivalent to `p to ~q`

Answer» Correct Answer - B
`(p to q) to {(~p to q) to q}`
`=(p to q) to {(p vvq) to q}`
`=( p to q) to {(~p to ~q) vvq}`
`=(p to q) to {(~p vvq) ^^(~qvvq)}`
`=(p to q) to (~p vvq)`
`=( pto q) to (p to q) `
=T
Discussion
14.

The negation of `~svv(~r^^s) ` is equivalent toA. `s^^~r`B. `s^^(r^^~s)`C. `svv(rvv~s)`D. `s^^r`

Answer» Correct Answer - D
`~svv(~r^^s)=(~svv~r)^^(~svvs)`
`=~(s^^r) ^^t`
`=~(s^^r) `
So, negation is `s^^r`.
Discussion
15.

The Boolean Expression `(p^^~ q)vvqvv(~ p^^q)`is equivalent to :(1) `~ p^^q`(2) `p^^q`(3) `pvvq`(4) `pvv~ q`A. `p^^q`B. `pvvq`C. `pvv~p`D. `~p^^q`

Answer» Correct Answer - B
`[(p^^~q)vvq](~qvvq)vv(~p^^q)`
`=(pvvq)^^(~qvvq)vv(~p^^q)`
`=(pvvq)^^[tvv(~p^^q)]`
`(pvvq)^^t`
`=pvvq`
Discussion
16.

`~(pvv(~pvvq))` is equal toA. `~p^^(p^^~q)`.B. `(pvv~q)v~p`C. none of theseD.

Answer» Correct Answer - A
`~[pvv(~pvvq)]`
`-=~p^^~(~pvvq)`
`-=~p^^(~(~p)^^~q)`
`-=~p^^(p^^~p)`
Discussion
17.

If the statements `(p^^~r) to (qvvr)`, q and r are all false, then pA. is trueB. is falseC. may be true or falseD. data is insufficient

Answer» Correct Answer - A
`(p^^~r) to (q vvr)` is false ,
Thus , `p^^~r` is true and `qvvr` is false .
Hence, `p ` must be true.
Discussion
18.

If p,q and r are simple propositions such that `(p^^q)^^(q^^r)` is true, thenA. p,q and r are all falseB. p,q and r are all trueC. p,qare true and r is falseD. p is true and q, r are false

Answer» Correct Answer - B
`(p^^q) ^^(q^^r)` is true which means that `p^^q` and `q^^r` are both true.
Therefore, p,q and r are all true,.
Discussion
19.

If p,q and r are simple propositions with truth values T,F and T , respectively, then the truth value of `(~pvvq) ^^~r to p` isA. 1B. FalseC. true if r is falseD. true if q is true

Answer» Correct Answer - A
`(~pvvq)^^ ~ r to p`
`-=(FvvF) ^^F to T`
`-=F to T`
`-=T`
Discussion