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7. Prove using truth table (A ⊕ B) ⊕ C = A ⊕ (B ⊕ C).
Ans. A B A ⊕ B C (A ⊕ B) ⊕ C B ⊕ C A ⊕ (B ⊕ C)
0 0 0 0 0 0 0
0 0 0 1 1 1 1
0 1 1 0 1 1 1
0 1 1 1 0 0 0
1 0 1 0 1 0 1
1 0 1 1 0 1 0
1 1 0 0 0 1 0
1 1 0 1 1 0 1

We find that the two columns are identical. Hence proved.

D. Higher order thinking skills (HOTS)
1. Why are NAND and NOR gates referred to as universal gates? Explain how these gates can be used to replicate AND or and NOT
gates.
Ans. NAND and NOR gates are called universal gates because any Boolean function can be implemented using just NAND or NOR gates.
This means we can create any other logic gate (AND or and NOT) using only one of these gates.
NAND as a universal gate:
• NOT gate can be implemented by connecting the same input to both inputs of a NAND gate: A NAND A = A'
• AND gate can be implemented by using two NAND gates in series. The output of the first NAND gate is then passed through
the second: (A NAND B) NAND (A NAND B) = A AND B
• OR gate can be implemented by inverting the inputs and then passing them through a NAND gate: (A NAND A) NAND (B NAND
B) = A OR B
NOR as a universal gate:
• NOT gate can be implemented by connecting the same input to both inputs of a NOR gate: A NOR A = A'
• OR gate can be implemented by using two NOR gates in series: (A NOR B) NOR (A NOR B) = A OR B
• AND gate can be implemented by inverting the inputs first and then passing them through a NOR gate: (A NOR A) NOR
(B NOR B) = A AND B
2. Construct a truth table for the compound proposition (A ∨ B) ∧ (∼C → A). Identify under which conditions the proposition
evaluates to true.
Ans. Truth table for the compound proposition (A ∨ B) ∧ (∼C → A):

A B C A ∨ B ∼C ∼C → A (A ∨ B) ∧ (∼C → A)
0 0 0 0 1 0 0
0 0 1 0 0 1 0
0 1 0 1 1 0 0
0 1 1 1 0 0 0
1 0 0 1 1 1 1
1 0 1 1 0 1 1
1 1 0 1 1 1 1
1 1 1 1 0 1 1
The proposition (A ∨ B) ∧ (∼C → A) evaluates to true under the following conditions:
• When A = 1 (true) and C = 1 (true) for both B = 0 and B = 1.
• When A=1(true), C=0(false) and B=0or B=1.
This is based on the truth table where the compound proposition evaluates to true for all instances where A=1, regardless of the
values of B or C.

Propositional Logic, Hardware Implementation, Arithmetic Operations 89