# Tutor profile: Rooha A.

## Questions

### Subject: Set Theory

Represent the following set in to Roaster form $$ A = \left \{ x: x \in N, x \leq 5\right \}$$

The set A is given in Set Builder form. It has been mentioned that the elements are natural numbers and are less than or equal to 5. The natural numbers starts from 1 and counts up. So as per the definition of the given set, the numbers are 1,2,3,4, and 5. The roaster form of set representation starts with a open brace, all the elements listed with comma as separation between them and ends with a close brace as shown below. $$ A = \left \{ 1,2,3,4,5\right \}$$ So we can represent the set $$ A = \left \{ x: x \in N, x \leq 5\right \} = \left \{ 1,2,3,4,5\right \}$$

### Subject: Electrical Engineering

Optimize the logical expression for the function $$ f(A, B) = \sum(0, 2)$$ using Boolean algebraic laws

The Boolean expression for the function can be written as $$f(A, B) = \sum (0, 2) = \overline{A}\cdot\overline{B} + A\cdot\overline{B} $$ We can see that the $$\overline{B}$$ is common in both product terms. So we can take $$\overline{B}$$ commonly outside of the bracket as given below. $$f(A, B) =\overline{B}\cdot(\overline{A}+A) \rightarrow (1)$$ We know that by complement rule of Boolean Algebra, the logical sum of a variable and it's complement is equal to logic 1 ($$\overline{A}+A = 1$$). So, we can modify the equation (1) as $$ f(A, B) =\overline{B}\cdot1 $$ $$ f(A, B) =\overline{B}$$ So the expression for the function f is optimized to $$ f(A, B) =\overline{B}$$.

### Subject: Algebra

Find the length and width of a rectangle whose perimeter is equal to 240 cm and its length is equal to twice its width.

Let us assume the length of the rectangle be 'L' and the width of the rectangle be 'W'. The perimeter of a rectangle can be calculated from the Length and width using the following equation $$Perimeter = 2\times Length + 2\times Width$$ If we substitute the 'L' for length and 'W' for width, we get $$Perimeter = 2\times L + 2\times W.$$ The perimeter is given and the value is 240 cm. Substituting the value in the above equation, we get $$2\times L + 2\times W = 240cm$$ In the above equation, all terms are even. so we can divide both sides of the equation with 2. we get, $$L + W = 120cm \rightarrow (1)$$ It has been given that the length is twice the width,.i.e. $$ L = 2\times W \rightarrow (2)$$ Substituting value for W from equation (2) into equation (1), we get, $$ 2 \times W + W = 120cm $$ $$ 3 \times W = 120cm$$ or $$ W = 120\div 3 $$ $$ W = 40 cm $$ Substituting the value for W into equation (2), we get $$ L = 2 \times 40cm $$ $$ L = 80 cm $$. So, the length of the rectangle is found to be 80 cm and the width of the rectangle is 40 cm.

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