Q.Add a pair of parentheses to each expression so that it evaluates to True.

Boolean Expressions: From Intuition to Precision

Imagine a light switch. It has exactly two states: on or off. There is no dimmer, no half‑on, no third option. A Boolean expression is just a way of describing conditions that also have only two possible outcomes: true or false.

That’s the entire intuition. Every Boolean expression you will ever write is a question that the computer (or a circuit) answers with a clear yes or no.

The Building Blocks

A Boolean expression is built from three things:

1. Boolean variables – these are placeholders that can hold either true (1) or false (0). For example, A, B, rain, isWeekend.
2. Logical operators – these combine variables into questions. There are three fundamental ones:
   • AND (written as $\cdot$ or $\land$): “Are both conditions true?”
   • OR (written as $+$ or $\lor$): “Is at least one condition true?”
   • NOT (written as $\overline{A}$ or $\lnot A$): “Is the condition false?”
3. Constants – true (1) and false (0).

How They Work: The Truth Table

The simplest way to see what a Boolean expression means is to list every possible combination of its variables and compute the result. That list is called a truth table.

Take the expression $F = A + B$ (A OR B). There are $2^2 = 4$ possibilities:

The result is 1 whenever at least one input is 1. That’s the OR operation.

Now take $G = A \cdot B$ (A AND B):

Only when both are 1 does the result become 1.

And NOT simply flips the value: $\overline{0} = 1$, $\overline{1} = 0$.

Why This Matters …