i) Deduce the Boolean expression for sum and carry, and then draw the logic circuit diagram. ii) Using…

Question

i) Deduce the Boolean expression for sum and carry, and then draw the logic circuit diagram. ii) Using Boolean identities, prove that XNOR gate is the complement of a XOR gate.

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d) i) Deduce the Boolean expression for sum and carry, and then draw the logic circuit diagram. From the truth table: A — B — Sum — Carry0 — 0 — 0 — 0 0 — 1 — 1 — 0 1 — 0 — 1 — 0 1 — 1 — 0 — 1 Step 1: Deduce the Boolean expression for Sum. The Sum output is 1 when A is 0 and B is 1, OR when A is 1 and B is 0. This corresponds to the XOR operation. Sum = A B Alternatively, in Sum of Products (SOP) form: Sum = AB + AB Step 2: Deduce the Boolean expression for Carry. The Carry output is 1 only when A is 1 AND B is 1. This corresponds to the AND operation. Carry = A · B Step 3: Draw the logic circuit diagram. The circuit consists of one XOR gate for the Sum output and one AND gate for the Carry output, with inputs A and B. A ---+--- XOR --- Sum | / | / B ---+-- | +--- AND --- Carry Note: Due to text-based limitations, the circuit diagram is represented using ASCII art. In a graphical representation, an XOR gate symbol and an AND gate symbol would be used. d) ii) Using Boolean identities, prove that XNOR gate is the complement of a XOR gate. Step 1: Write the Boolean expression for an XOR gate. The XOR operation (A B) is defined as: A B = AB + AB Step 2: Find the complement of the XOR expression. We need to prove that A B = A B. Let's take the complement of the XOR expression: A B = AB + AB; Step 3: Apply De Morgan's Theorem (X+Y = XY). AB + AB = (AB) · (AB); Step 4: Apply De Morgan's Theorem again (XY = X+Y). = (A + B) · (A + B); Step 5: Simplify using the double complement rule (X = X). = (A + B) · (A + B) Step 6: Expand the expression (distributive law). = AA + AB + BA + BB; Step 7: Apply Boolean identities (XX = 0 and X · 0 = 0). = 0 + AB + AB + 0 Step 8: Simplify the expression. = AB + AB Step 9: Recognize the resulting expression as the XNOR gate. The expression AB + AB is the definition of the XNOR operation (A B). Therefore, we have shown that: A B = A B This proves that the XNOR gate is the complement of an XOR gate.

I) Deduce the Boolean expression for sum and carry, and then draw the logic circuit diagram. ii) Using Boolean identities, prove that XNOR gate is the complement of a XOR gate.

Step-by-step solution

Deduce the Boolean expression for Sum.

Deduce the Boolean expression for Carry.

Draw the logic circuit diagram.

Write the Boolean expression for an XOR gate.

Find the complement of the XOR expression.

Apply De Morgan's Theorem ( $\overline{X+Y} = \bar{X}\bar{Y}$ ).

Apply De Morgan's Theorem again ( $\overline{XY} = \bar{X}+\bar{Y}$ ).

Simplify using the double complement rule ( $\overline{\bar{X}} = X$ ).

Expand the expression (distributive law).

Apply Boolean identities ( $X\bar{X} = 0$ and $X \cdot 0 = 0$ ).

Simplify the expression.

Recognize the resulting expression as the XNOR gate.

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