Question

Assume that \(a=1, b=2\), and \(c=2\). What is the value of each of the following Boolean expressions? a. \((a>1)\) OR \((b=c)\) b. \([(a+b)>c]\) AND \((b

Short answer

a. True b. False c. False d. False e. False

Step-by-step solution

Evaluate (a>1) OR (b=c)

Substitute the values into each subexpression:- For \(a > 1\), substitute \(a = 1\). This becomes \(1 > 1\) which is False.- For \(b = c\), substitute \(b = 2\) and \(c = 2\). This becomes \(2 = 2\) which is True.The expression \((a > 1)\) OR \((b = c)\) is evaluated as:- False OR True, which simplifies to True because OR requires only one operand to be true.

Evaluate [(a+b)>c] AND (b

Substitute the values:- For \((a + b) > c\), substitute \(a = 1\), \(b = 2\), and \(c = 2\). It becomes \((1 + 2) > 2\) which evaluates to \(3 > 2\), True.- For \(b < c\), substitute \(b = 2\), \(c = 2\). It becomes \(2 < 2\) which evaluates to False.The expression \([(a + b) > c]\) AND \((b < c)\) evaluates as:- True AND False, which results in False because AND requires both operands to be true.

Evaluate NOT(a=1)

Substitute the value into the subexpression:- For \(a = 1\), substitute \(a = 1\). It becomes \(1 = 1\) which is True.The expression \(\operatorname{NOT}(a = 1)\) is:- NOT True, which results in False because NOT negates the operand.

Evaluate NOT[(a=b) OR (b=c)]

Substitute the values:- For \(a = b\), substitute \(a = 1\), \(b = 2\). It becomes \(1 = 2\), which is False.- For \(b = c\), substitute \(b = 2\), \(c = 2\). It becomes \(2 = 2\), which is True.The expression \((a = b)\) OR \((b = c)\) becomes False OR True, simplifying to True.Thus, \(\operatorname{NOT}[(a = b)\) OR \((b = c)]\) is NOT True, which results in False.

Evaluate (a=1) AND (b=1) AND (c=2)

Substitute the values:- For \(a = 1\), substitute \(a = 1\). It is True.- For \(b = 1\), substitute \(b = 2\). It is False since 2 does not equal 1.- For \(c = 2\), substitute \(c = 2\). It is True.The expression \((a = 1)\) AND \((b = 1)\) AND \((c = 2)\) evaluates to True AND False AND True. Since AND requires all operands to be true, the final result is False.

Key concepts

Boolean Expressions

Boolean expressions are statements that evaluate to either true or false. In the world of Boolean Algebra, these expressions form the foundation for logical computation. Think of a Boolean expression like a question that can have only two possible answers: yes (true) or no (false).

Boolean expressions often consist of variables and operators. In our exercise, variables are represented by letters like \(a\), \(b\), and \(c\), and they hold specific values. Operators are logical symbols like AND, OR, and NOT, which determine how the statements are evaluated.

Consider this example:

• The expression \((a > 1)\) OR \((b = c)\) uses the greater-than and equality operators respectively. Here, \((a > 1)\) checks if \(a\) is greater than 1, while \((b = c)\) checks if \(b\) equals \(c\). The OR operator connects them, resulting in true if at least one condition is true.

The beauty of Boolean expressions lies in their ability to simplify complex logical decisions into true or false values.

Logical Operations

Logical operations are crucial in evaluating Boolean expressions. They act as the building blocks that shape how Boolean expressions are evaluated. The primary logical operations are AND, OR, and NOT.

• **AND Operation**: This operation returns true only if all conditions or subexpressions involved are true. In Boolean algebra, "AND" is symbolized by multiplication. For instance, \(((a+b) > c) \) AND \((b < c)\) evaluates to false if even one of the conditions is false.


• **OR Operation**: This operation yields true if at least one of the conditions is true. In Boolean algebra, it resembles addition. Consider the expression \((a > 1)\) OR \((b = c)\)—it evaluates to true if either condition is met.


• **NOT Operation**: Distinct from AND and OR, the NOT operation negates or reverses the truth value of a condition. If a condition is true, applying a NOT before it makes it false. For example, \(\operatorname{NOT}(a = 1)\) negates the truth of \(a = 1\), as demonstrated in your problem.

Combining these operations allows for a diverse range of logical problem-solving possibilities.

Truth Tables

Truth tables are a visual representation tool used to understand the outcome of logical expressions based on all possible input values. They list all possible combinations of truth values for the variables involved and show the result of the expression for each combination.

Creating a truth table involves:

• Listing every possible value of the variables (e.g., true or false, 1 or 0).
• Evaluating the Boolean expression for each of these value combinations.
• Recording whether the entire expression is true or false for each combination.

Let's say you have two variables, \(p\) and \(q\), and the expression \(p\) AND \(q\). A truth table would show results for:- When both \(p\) and \(q\) are true, the expression is true.- When \(p\) is true and \(q\) is false, or vice versa, the expression is false.- When both \(p\) and \(q\) are false, the expression is also false.

Truth tables are an excellent way to systematically verify the result of logical statements, providing clear insights into how different inputs affect the overall evaluation.