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A logical binary relation $\odot$, is defined as follows: 

$$\begin{array}{|l|l|l|} \hline \textbf{A} & \textbf{B}& \textbf{A} \odot \textbf{B}\\\hline \text{True} & \text{True}& \text{True}\\\hline \text{True} & \text{False}& \text{True}\\\hline \text{False} & \text{True}& \text{False}\\\hline \text{False} & \text{False}& \text{True}\\\hline   \end{array}$$

Let $\sim$ be the unary negation (NOT) operator, with higher precedence then $\odot$.

Which one of the following is equivalent to $A\wedge B$ ?

  1. $(\sim A\odot B)$
  2. $\sim(A \odot \sim B)$
  3. $\sim(\sim A\odot\sim B)$
  4. $\sim(\sim A\odot B)$
in Set Theory & Algebra by Active (3.3k points)
edited by | 1.5k views
0
plz someone answer it, using truth table method
0
Oher way

Use kmap to obtain expression, then try to match with options.
0
We can also used truth table bcz they have only two variables

7 Answers

+35 votes
Best answer

This question is easier to answer With Boolean Algebra.

$A\odot B \equiv B\to A$, i.e. $(\neg B\vee A).$

Now, lets look at Option D

$\neg (\neg A\odot B)$
$\quad \equiv \neg (B\to \neg A)$
$\quad \equiv \neg (\neg B\vee \neg A)$
$\quad \equiv B\wedge A$

So Answer is D.

Other options:

  1. $\neg B\vee \neg A$
  2. $\neg B\wedge \neg A$
  3. $\neg B\wedge  A$
by Boss (41.5k points)
edited by
+9 votes
Instead of checking all the answers, we can approach similar problems by finding out the relationship between the given operation and the one which we have to express.

Here,
\begin{align} A\odot B &= AB+A\bar B+\bar A \bar B\\ &= A+\bar B \\ &=\overline{\bar{A}B}  \end{align} $$ \overline{A\odot B} = \bar AB \implies AB = \color{red}{\overline{\bar A \odot B}}$$
by Loyal (5.4k points)
edited by
0
nice thinking ....
0
How did you get this middle term?

AB(BAR)
+6 votes

This truth table is of B->A ie B' V A

so operator given in question is implication

we want A ∧B

which is B'->A'

so ans is d

by Boss (31.2k points)
0
@pooja, Option b = ~b ∧ ~a.
0
Similar que was asked In gate2009 it was asked for or there...i have answered it for gate 2009 i think answer was uploaded at wrong place..sry for that yes ans should be d for this que
+2 votes

therefore option D is correct

by (49 points)
+2 votes
we can also Answer this question by putting TRUE for both A and B, and the result should be TRUE for AND Operation,

By this method, only D will qualify.
by Active (3k points)
0 votes
A B ~A ~B (~A op B) ~(A op ~B) ~(~A op ~B) ~(~A op B)   A ⋀ B
T T F F F F F T T
T F F T T F T F F
F T T F T F F F F
F F T T T T F F F

op is the binary logical realtion mentioned in the question.

So second last column i.e. option D matches AND operation of A,B.

ago by (23 points)
–1 vote
Option d
by Active (3.8k points)
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