Question

(De Morgan’s Laws) In this chapter, we have discussed the logical operators &&, &, ||, |, ^ and !. De Morgan’s Laws can sometimes make it more convenient for us to express a logical expression. These laws state that the expression !(condition1 && condition2) is logically equivalent to the expression (!condition1 || !condition2). Also, the expression !(condition1 || condition2) is logically equivalent to the expression (!condition1 && !condition2). Use De Morgan’s Laws to write equivalent expressions for each of the following, then write an application to show that both the original expression and the new expression in each case produce the same value: a) !( x < 5 ) && !( y >= 7 ) b) !( a == b ) || !( g != 5 ) c) !( ( x <= 8 ) && ( y > 4 ) ) d) !( ( i > 4 ) || ( j <= 6 ) )

Short answer

After applying De Morgan's Laws, equivalent expressions are found. Verification confirms equivalence.

Step-by-step solution

Understand De Morgan's Laws

De Morgan's Laws provide a way to simplify logical expressions. They state that 1. The negation of a conjunction is the disjunction of the negations: \( eg (P \land Q) \equiv eg P \lor eg Q \)2. The negation of a disjunction is the conjunction of the negations: \( eg (P \lor Q) \equiv eg P \land eg Q \). We will use these for rewriting expressions.

Simplify Expression (a)

Given: \(!( x < 5 ) \land !( y \geq 7 )\). Use De Morgan's Law on each part:- \(!( x < 5 )\) is equivalent to \((x \geq 5)\) because the negation of 'less than' is 'greater than or equal to'.- \(!( y \geq 7 )\) is equivalent to \((y < 7)\).So, the expression is \((x \geq 5) \land (y < 7)\).

Simplify Expression (b)

Given: \(!( a == b ) \lor !( g eq 5 )\). Use De Morgan's Law on each part:- \(!( a == b )\) is equivalent to \((a eq b)\).- \(!( g eq 5 )\) is equivalent to \((g == 5)\).Thus, the expression is \((a eq b) \lor (g == 5)\).

Simplify Expression (c)

Given: \(!( ( x \leq 8 ) \land ( y > 4 ) )\). Use De Morgan's Law:- For the entire expression, apply the negation: \(eg(( x \leq 8 ) \land ( y > 4 ))\) becomes \(eg(x \leq 8) \lor eg(y > 4)\).- \(eg(x \leq 8)\) simplifies to \((x > 8)\).- \(eg(y > 4)\) simplifies to \((y \leq 4)\).The simplified expression is \((x > 8) \lor (y \leq 4)\).

Simplify Expression (d)

Given: \(!( ( i > 4 ) \lor ( j \leq 6 ) )\). Use De Morgan's Law:- For the entire expression, apply the negation: \(eg(( i > 4 ) \lor ( j \leq 6 ))\) becomes \(eg(i > 4) \land eg(j \leq 6)\).- \(eg(i > 4)\) simplifies to \((i \leq 4)\).- \(eg(j \leq 6)\) simplifies to \((j > 6)\).The simplified expression is \((i \leq 4) \land (j > 6)\).

Verify Equivalence through Application

To verify the equivalence, write a program or logical application for each original and simplified expression, showing that they evaluate to the same values when substituting different values for the variables. Compare results in cases like: - For expression (a): Check with \(x = 4, y = 8\); should return True for both.- For expression (b): Check with \(a = 2, b = 3, g = 5\); should return True for both.- For expression (c): Check with \(x = 10, y = 3\); both should return True.- For expression (d): Check with \(i = 2, j = 8\); both should return True.

Key concepts

Logical Operators

Logical operators are special symbols we use to connect or modify boolean expressions, which yield either "true" or "false". In programming and logic, we typically use operators such as `&&` for AND, `||` for OR, and `!` for NOT. These operators let us formulate complex conditions in a straightforward way.
De Morgan's Laws play a crucial role in manipulating these operators to optimize and simplify logical expressions. For example, you can replace complex conditionals using combinations of these logical operators and even simplify them by appropriately applying De Morgan's Laws.
The power of logical operators is their ability to represent an infinite number of logical conditions using a finite set of rules. Let's break them down a bit more:

• `&&` (AND) - returns true if both conditions are true. Think of it as needing both statements to be correct for the overall expression to be true.
• `||` (OR) - is true if at least one of the conditions is true. This operator offers flexibility when working with multiple scenarios.
• `!` (NOT) - negates a condition, i.e., it reverses the truth value of an expression.

De Morgan's Laws, hence, give us new perspectives to rearrange these logical operators to potentially make them simpler for humans and machines alike.

Boolean Expressions

A boolean expression is a statement that can be either true or false. These expressions are fundamental in controlling flow within a program, as they help in decision-making processes.
For example, you might have a boolean expression that checks if a user has reached a certain age, or if a stock level is low. These checks either lead to intended actions or logical pathways in code. With boolean expressions, you can: - Compare different data values using comparison operators (>, <, ==, etc.)
- Use logical operators (&&, ||, !) to combine multiple conditions.

In the realm of De Morgan's Laws, boolean expressions are pivotal as they are what we seek to transform for efficiency and clarity. For instance, instead of writing a compound boolean expression as `!(x < 5) && !(y >= 7)`, we can simplify it using De Morgan's Laws to `x >= 5 && y < 7`, making it clearer and potentially easier for others to comprehend. These expressions thus enable simple to complex evaluations in both everyday decision making and formal program structures.

Logical Equivalence

Logical equivalence refers to two or more expressions yielding the same result in any possible scenario. It's a crucial concept in logical operations and programming, as it allows us to transform expressions without changing their original meaning.
De Morgan's Laws are utilized to establish logical equivalence in expressions involving logical operators. By rewriting expressions using these laws, we ensure that what might appear differently is actually the same at its core.
Consider the expressions `!(x < 5) && !(y >= 7)` and `x >= 5 && y < 7`. While structurally different, they are logically equivalent because they evaluate to the same boolean values for any given values of `x` and `y`.
Understanding logical equivalence is vital for debugging, optimizing, and verifying code, since it guards against potential logical errors in program logic.

Program Verification

Program verification is a process ensuring that a program behaves as intended under all possible conditions and inputs. With this, programmers validate that the logic within the code is correct and performs the desired tasks effectively without any errors.
Effective program verification involves using logical operators, boolean expressions, and logical equivalence. For instance, in verifying that two expressions are equivalent, you test various input conditions to check their outcomes. De Morgan’s Laws facilitate this by allowing programmers to initially test original versus transformed expressions in programs.
To efficiently verify expressions like `!(i > 4) || !(j <= 6)` and ensure logical equivalence with `i <= 4 && j > 6`, you might break it down into systematic tests:

• Test different values for `i` and `j` to confirm that the evaluations are consistent across both expressions.
• Document the outcomes to ensure full understanding and accuracy on why and how these expressions work.

Thus, program verification ensures reliability and correctness, reinforcing the robustness of applications dealing with various logical conditions.