Law of Logical Equivalence in Discrete Mathematics

Suppose there are two compound statements, X and Y, which will be known as logical equivalence if and only if the truth table of both of them contains the same truth values in their columns. With the help of symbol = or ⇔, we can represent the logical equivalence. So X = Y or X ⇔ Y will be the logical equivalence of these statements.

With the help of the logical equivalence definition, we have cleared that if the compound statements X and Y are logical equivalence, in this case, the X ⇔ Y must be Tautology.

Laws of Logical Equivalence

In this law, we will use the 'AND' and 'OR' symbols to explain the law of logical equivalence. Here, AND is indicated with the help of ∧ symbol and OR is indicated with the help of ∨ symbol. There are various laws of logical equivalence, which are described as follows:

Idempotent Law:

In the idempotent law, we only use a single statement. According to this law, if we combine two same statements with the symbol ∧(and) and ∨(or), then the resultant statement will be the statement itself. Suppose there is a compound statement P. The following notation is used to indicate the idempotent law:

The truth table for this law is described as follows:

This table contains the same truth values in the columns of P, P ∨ P and P ∧ P.

Hence we can say that P ∨ P = P and P ∧ P = P.

Commutative Laws:

The two statements are used to show the commutative law. According to this law, if we combine two statements with the symbol ∧(and) or ∨(or), then the resultant statement will be the same even if we change the position of the statements. Suppose there are two statements, P and Q. The proposition of these statements will be false when both statements P and Q are false. In all the other cases, it will be true. The following notation is used to indicate the commutative law:

The truth table for these notations is described as follows:

This table contains the same truth values in the columns of P ∨ Q and Q ∨ P.

Hence we can say that P ∨ Q ? Q ∨ P.

Same as we can prove P ∧ Q ? Q ∧ P.

Associative Law:

The three statements are used to show the associative law. According to this law, if we combine three statements with the help of brackets by the symbol ∧(and) or ∨(or), then the resultant statement will be the same even if we change the order of brackets. That means this law is independent of grouping or association. Suppose there are three statements P, Q and R. The proposition of these statements will be false when P, Q and R are false. In all the other cases, it will be true. The following notation is used to indicate the associative law:

The truth table for these notations is described as follows:

This table contains the same truth values in the columns of P ∨ (Q ∨ R) and (P ∨ Q) ∨ R.

Hence we can say that P ∨ (Q ∨ R) ? (P ∨ Q) ∨ R.

Same as we can prove P ∧ (Q ∧ R) ? (P ∧ Q) ∧ R

Distributive Law:

The three statements are used to show the distributive law. According to this law, if we combine a statement by the ∨(OR) symbol with the two other statements which are joined with the symbol ∧(AND), then the resultant statement will be the same even if we are separately combining the statements with the symbol ∨(OR) and combining the joined statements with ∧(AND). Suppose there are three statements P, Q and R. The following notation is used to indicate the distributive law:

P ∨ (Q ∧ R) ? (P ∨ Q) ∧ (P ∨ R)

P ∧ (Q ∨ R) ? (P ∧ Q) ∨ (P ∧ R)

The truth table for these notations is described as follows:

This table contains the same truth values in the columns of P ∨ (Q ∧ R) and (P ∨ Q) ∧ (P ∨ R).

Hence we can say that P ∨ (Q ∧ R) = (P ∨ Q) ∧ (P ∨ R)

Same as we can prove P ∧ (Q ∨ R) ? (P ∧ Q) ∨ (P ∧ R)

Identity Law:

A single statement is used to show the identity law. According to this law, if we combine a statement and a True value with the symbol ∨(or), then it will generate the True value. If we combine a statement and a False value with the symbol ∧(and), then it will generate the statement itself. Similarly, we will do this with the opposite symbols. That means if we combine a statement and a True value with the symbol ∧(and), then it will generate the statement itself, and if we combine a statement and a False value with the symbol ∨(or), then it will generate the False value. Suppose there is a compound statement P, a true value T and a false value F. The following notation is used to indicate the identity law:

The truth table for these notations is described as follows:

This table contains the same truth values in the columns of P ∨ T and T. Hence, we can say that P ∨ T = T. Similarly, this table also contains the same truth values in the columns of P ∨ F and P. Hence we can say that P ∨ F = P.

Same as we can prove P ∧ T ? P and P ∧ F ? F

Complement Law:

A Single statement is used in the complement law. According to this law, if we combine a statement with its complement statement with the symbol ∨(or), then it will generate the True value, and if we combine these statements with the symbol ∧(and), then it will generate the False value. If we negate a true value, then it will generate a false value, and if we negate a false value, then it will generate the true value.

The following notation is used to indicate the complement law:

The truth table for these notations is described as follows:

This table contains the same truth values in the columns of P ∨ ¬P and T. Hence, we can say that P ∨ ¬P = T. Similarly, this table also contains the same truth values in the columns of P ∧ ¬P and F. Hence we can say that P ∧ ¬P = F.

This table contains the same truth values in the columns of ¬T and F. Hence, we can say that ¬T = F. Similarly, this table contains the same truth values in the columns of ¬F and T. Hence we can say that ¬F = T.

Double Negation Law or Involution Law

A single statement is used to show the double negation law. According to this law, if we do the negation of a negated statement, then the resultant statement will be the statement itself. Suppose there is a statement P and a negate statement ¬P. The following notation is used to indicate the Double negation law:

The truth table for these notations is described as follows:

This table contains the same truth values in the columns of ¬(¬P) and P. Hence we can say that ¬(¬P) = P.

De Morgan's Law:

The two statements are used to show De Morgan's law. According to this law, if we combine two statements with the symbol ∧(AND) and then do the negation of these combined statements, then the resultant statement will be the same even if we combine the negation of both statements separately with the symbol ∨(OR). Suppose there are two compound statements, P and Q. The following notation is used to indicate De Morgan's Law:

The truth table for these notations is described as follows:

This table contains the same truth values in the columns of ¬(P ∧ Q) and ¬ P ∨ ¬Q. Hence we can say that ¬(P ∧ Q) = ¬ P ∨ ¬Q.

Same as we can prove ¬(P ∨ Q) ? ¬P ∧ ¬Q

Absorption Law:

The two statements are used to show the absorption law. According to this law, if we combine a statement P by ∨(OR) symbol with the same statement P and one other statement Q, which are joined with the symbol ∧(AND), then the resultant statement will be the first statement P. The same result will be generated if we interchange the symbols. Suppose there are two compound statements, P and Q. The following notation is used to indicate the Absorption Law:

The truth table for these notations is described as follows:

This table contains the same truth values in the columns of P ∨ (P ∧ Q) and P. Hence we can say that P ∨ (P ∧ Q) ? P.

Similarly, this table also contains the same truth values in the columns of P ∧ (P ∨ Q) and P. Hence we can say that P ∧ (P ∨ Q) ? P.

Examples of Logical Equivalence

There are various examples of logical equivalence. Some of them are described as follows:

Example 1: In this example, we will establish the equivalence property for a statement, which is described as follows:

p → q ? ¬p ∨ q

Solution:

We will prove this with the help of a truth table, which is described as follows:

This table contains the same truth values in the columns of p → q and ¬p ∨ q. Hence we can say that p → q ? ¬p ∨ q.

Example 2: In this example, we will establish the equivalence property for a statement, which is described as follows:

P ↔ Q ? (P → Q) ∧ (Q → P)

Solution:

This table contains the same truth values in the columns of P ↔ Q and (P → Q) ∧ (Q → P). Hence we can say that P ↔ Q ? (P → Q) ∧ (Q → P).

Example 3: In this example, we will use the equivalent property to prove the following statement:

p ↔ q ? ( p ∧ q ) ∨ ( ¬ p ∧ ¬q)

Solution:

To prove this, we will use some of the above-described laws and from this law we have:

p ↔ q ? (¬p ∨ q) ∧ (¬q ∨ p) ...........(1)

Now we will use the Commutative law in the above equation and get the following:

? (¬ p ∨ q) ∧ (p ∨ ¬q)

Now we will use the Distributive law in this equation and get the following:

? (¬ p ∧ (p ∨ ¬q)) ∨ (q ∧ (p ∨ ¬q))

Now we will use Distributive law in this equation and get the following:

? (¬ p ∧ p) ∨ (¬p ∧ ¬q) ∨ (q ∧ p) ∨ (q ∧ ¬q)

Now we will use the complement law in this equation and get the following:

? F ∨ (¬p ∧ ¬q) ∨ (q ∧ p) ∨ F

Now we will use the identity law and get the following:

? (¬ p ∧ ¬ q) ∨ (q ∧ p)

Now we will use the Commutative law in this equation and get the following:

? (p ∧ q) ∨ (¬ p ¬q)

Finally, equation (1) becomes the following:

p ↔ q ? (p ∧ q) ∨ (¬ p ¬q)

Finally, we can say that the equation (1) becomes p ↔ q ? (p ∧ q) ∨ (¬ p ∧ ¬q)