GO Classes CS Test Series 2025 | Discrete Mathematics | Topic Wise Test 1 | Question: 9

Answer 1

p:q?r $\equiv$ If p then q  and if not p then r $\equiv$ $(p\rightarrow q)\wedge (\neg p\rightarrow r)$

A) Probability of expression, p:q?r to be true = $\frac{I}{J}$ = $\frac{4}{8}$ = $\frac{1}{2}$
I = Total no. of rows in the truth table where given expression is true = #Models
J = Total no. of rows in the truth table = #Interpretaions

B) $(p\rightarrow p)\wedge (\neg p\rightarrow p) \equiv p$ (Contingency)

C) $(p\rightarrow p)\wedge (\neg p\rightarrow \neg p) \equiv True$ (Tautology)

D) $(\neg p\rightarrow p)\wedge (p\rightarrow \neg p) \equiv False$ (Contradiction)

Correct Option: A, C

 

Comments

Nicely answered. Students can also watch: Detailed Video Solution (Click Here) 

Answer 2

• A. p?p:p

           There are two mutually exclusive cases here. We can translate as follows

           =  $(p\Lambda p )V({p}'\Lambda p)$

            P can take two values true and false. So, probability of p = probability of $p{}'$ = ½. 

            So, total probability is $\left ( \left ( 1/2 * 1/2 \right ) \right ) + \left ( \left ( 1/2 * 1/2 \right ) \right )$

            = 1/2

 

• A. p?p:p

           = $\left ( p \Lambda p\right ) V \left ( p{}' \Lambda p \right )$

           = $pVF = p$ ( not a tautology )

100. p?p:$p{}'$
     
     = $\left (p  \Lambda p \right ) V \left ( p{}' \Lambda p{}'\right ) = p V p{}' = T$ ( tautology )

500. $p{}' ? p:p{}'
     
     = \left ( p{}' \Lambda p \right ) V \left ( p \Lambda p{}' \right )
     
     = F V F = F$ ( not a tautology )
     
     So, correct options are A , C