Bitwise Operations: Concepts, Formulas, and Clear Examples

Technical aptitude tests bitwise logic on integers at the bit level—AND, OR, XOR, NOT, left/right shifts—without running code. Know truth tables and how shifts multiply/divide by powers of two.

Questions ask for output of expressions, count set bits, or detect tricks ($n \& (n-1)$ clears lowest set bit). Think in binary for small n; use algebraic identities for speed on MCQs.

Truth tables

AND

$$0\&0=0,\ 1\&1=1,\ \text{else } 0$$

OR

$$0|0=0,\ \text{any 1 } \Rightarrow 1$$

XOR

$$a \oplus b = 1 \text{ iff exactly one of } a,b \text{ is } 1$$

NOT

$$\sim 0 = 1,\ \sim 1 = 0 \text{ (within bit width)}$$

Shifts

Left shift

$$x << k = x \times 2^k \text{ (unsigned; overflow wraps mod } 2^w)$$

Right shift

$$x >> k = \lfloor x / 2^k \rfloor \text{ for non-negative } x$$

Example — Evaluate 5 & 3

5 = 101, 3 = 011 → AND = 001 = 1.

Useful identities

• $a \oplus a = 0$; $a \oplus 0 = a$.
• $a \& (a-1)$ removes lowest 1-bit.
• Check even/odd: $n \& 1$.

Masking and flags

Set bit k: $n | (1 << k)$. Clear bit k: $n \& \sim(1 << k)$. Toggle: $n \oplus (1 << k)$.

Signed vs unsigned (concept)

• Right shift of negative numbers may sign-extend—read question’s machine model.

Exam pitfalls

• Operator precedence: shift before additive in C-like languages.
• Width: 8-bit result vs 32-bit promotion.

Step-by-step method for exam questions

1. Convert operands to binary (8–16 bits enough).
2. Apply operator row-wise per bit or use identities.
3. For shifts, compute power-of-two multiply/divide.
4. Watch unsigned vs signed if stated.
5. Convert result back to decimal.
6. Sanity-check magnitude (AND ≤ min operands).

Formula cheat sheet

Frequently asked questions

Do I need to know two's complement deeply?

Basics for NOT width and signed >>; most items use small positive integers.

Language-specific?

Follow C/Java rules unless question specifies Python (& on int).

Fast parity?

XOR all bits or use $n \oplus (n>>1) \oplus …$ pattern on paper for tiny n.

Practice on this platform

Use 4 practice sets (10 MCQs, detailed solution after each answer) and 2 timed mocks (20 minutes) on this topic to convert formulas into speed and accuracy.