Grade 9

Intermediate

10 min

Lesson 11 of 12

Mapping Logic: Truth Tables

Not Started

Learn to use truth tables to visualize and solve complex boolean expressions.

Ever wondered how a computer 'decides' to fire a pixel or save a file? It isn't magic—it's a map of every possible 'Yes' or 'No' answer, organized into a powerful tool called a Truth Table.

Learning Objectives

1.By the end of this lesson, you will be able to construct a complete truth table for a 2-input logic circuit.
2.You will evaluate boolean expressions containing multiple operators using the correct order of operations.
3.You will analyze a truth table to identify if a complex circuit can be simplified into a single gate.

The Blueprint of Logic

A Truth Table is a mathematical table used to show the output of a logic circuit for every possible combination of inputs. In computer science, we use $1$ to represent TRUE (on) and $0$ to represent FALSE (off). For a circuit with $n$ inputs, there are always $2^n$ possible combinations. For a standard 2-input circuit (Inputs $A$ and $B$ ), we have $2^2 = 4$ rows. This ensures we never miss a scenario, from 'both off' to 'both on.' Think of it as a master checklist for a machine's brain.

Let's map the simplest 2-input gate: the AND gate ( $Q = A \cdot B$ ). 1. List inputs $A$ and $B$ in binary order: $(0,0), (0,1), (1,0), (1,1)$ . 2. Apply the rule: The output $Q$ is $1$ ONLY if both $A$ and $B$ are $1$ . 3. Resulting Output Column: $0, 0, 0, 1$ .

Quick Check

If a logic circuit has 3 inputs (A, B, and C), how many rows will the truth table require to cover all combinations?

Use the formula

2^n

where

n

is the number of inputs.

Answer

8 rows

The Hierarchy of Operations

When expressions get complex, like $Q = \neg(A \lor B)$ , we must follow a specific order, similar to PEMDAS in math. In Boolean Algebra, the order is: NOT ( $eg$ ) first, then AND ( $\land$ ), and finally OR ( $\lor$ ). If there are parentheses, always solve the logic inside them first. To build a table for a complex expression, we create 'intermediate' columns to track each step of the calculation before reaching the final output.

Evaluate $Q = \neg(A \land B)$ : 1. Create columns for $A$ , $B$ , and the intermediate step $(A \land B)$ . 2. Fill $(A \land B)$ : $0, 0, 0, 1$ . 3. Apply the NOT gate to that column to get $Q$ . 4. Final Output $Q$ : $1, 1, 1, 0$ . This is also known as a NAND gate!

Quick Check

In the expression $Q = A \lor \neg B$ , which operation do you perform first?

Remember the hierarchy: NOT comes before OR.

Answer

The NOT operation (\neg B)

Simplifying the Circuit

Engineers use truth tables to see if a circuit is over-complicated. If two different boolean expressions produce the exact same output column in a truth table, they are logically equivalent. For example, a circuit with ten gates that results in the output $0, 1, 1, 1$ is just a fancy (and expensive) OR gate. Simplifying circuits reduces heat, saves power, and makes computers faster.

Analyze $Q = (A \land B) \lor (A \land \neg B)$ : 1. Calculate $(A \land B)$ : $0, 0, 0, 1$ . 2. Calculate $\neg B$ : $1, 0, 1, 0$ . 3. Calculate $(A \land \neg B)$ : $0, 0, 1, 0$ . 4. OR the results from step 1 and step 3: $0, 0, 1, 1$ . 5. Notice the output $0, 0, 1, 1$ is identical to Input $A$ . The entire circuit simplifies to just $Q = A$ !

Key Takeaways

1A truth table must account for $2^n$ combinations of inputs.
2The Boolean order of operations is NOT, then AND, then OR.
3Intermediate columns are essential for solving multi-step expressions without errors.
4Logical equivalence allows engineers to replace complex circuits with simpler, identical ones.

Quick Check

Multiple Choice

What is the output of an OR gate when the inputs are $A=1$ and $B=0$ ?

Multiple Choice

Which column represents the intermediate step for $Q = \neg A \land B$ ?

True/False

If two different boolean expressions have the same truth table output, they are logically equivalent.

What's Next?

Review Tomorrow

In 24 hours, try to draw the truth table for a XOR gate (Exclusive OR), where the output is 1 only if the inputs are different.

Practice Activity

Try this on your own: Create a truth table for $Q = \neg(A \lor B)$ and compare it to the table for $Q = \neg A \land \neg B$ . Are they the same?

Browse

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Mapping Logic: Truth Tables

Learning Objectives

The Blueprint of Logic

The Hierarchy of Operations

Simplifying the Circuit

Key Takeaways

Quick Check

What's Next?

Mapping Logic: Truth Tables

Learning Objectives

The Blueprint of Logic

The Hierarchy of Operations

Simplifying the Circuit

Key Takeaways

Quick Check

What's Next?