# Theory of Computation: Question Set – 06

#### What is the implication of two propositions?

The implication of two propositions is a proposition that is false only when the antecedent is true and the consequent is false.

#### What is the biconditional of two propositions?

The biconditional of two propositions is a proposition that is true only when both of the original propositions have the same truth value.

#### What is a proof in mathematics?

A proof is a logical argument that establishes the truth of a mathematical statement or proposition. In a proof, each step must follow logically from the previous steps, starting from a set of axioms or assumptions and leading to the desired conclusion.

#### What are the elements of a mathematical proof?

A mathematical proof typically consists of the following elements:

• Statement of the theorem or proposition to be proved.
• Introduction of any relevant definitions or concepts.
• Listing of any given information or assumptions.
• Development of the argument or proof, consisting of a sequence of logical steps leading from the given information and assumptions to the desired conclusion. Conclusion or summary of the proof, stating the theorem or proposition that has been proved.

#### What are some common proof techniques used in mathematics?

There are many proof techniques used in mathematics, including:

• Direct proof: This involves starting with the given information and proceeding through a series of logical steps to arrive at the desired conclusion.
• Induction: This involves proving a statement for a base case, then assuming it is true for some general case, and using that assumption to prove the statement for the next case.
• Contradiction: This involves assuming that the statement to be proved is false, and then deriving a contradiction or inconsistency from that assumption.
• Counterexample: This involves providing a specific example that contradicts the statement to be proved. Reduction to absurdity: This involves showing that the assumption of the negation of the statement to be proved leads to a logical absurdity.

#### What is a proof by contradiction?

A proof by contradiction is a technique used in mathematics where one assumes the negation of the statement to be proved and shows that this leads to a contradiction or inconsistency. Since a contradiction cannot be true, this implies that the original statement must be true.

#### What is mathematical induction?

Mathematical induction is a proof technique used to prove statements that depend on a variable that takes on discrete, integer values, such as natural numbers. The technique involves proving a base case for the smallest possible value of the variable, and then assuming that the statement is true for some general value of the variable, and using that assumption to prove the statement for the next larger value of the variable. This process is continued until the statement has been proved for all values of the variable.

#### What is a counterexample in mathematics?

A counterexample is a specific example that contradicts a mathematical statement or proposition. A counterexample can be used to show that a statement is false, or to provide insight into why a statement is true or false.

#### What is a proof by induction?

A proof by induction is a technique used to prove statements that depend on a variable that takes on discrete, integer values, such as natural numbers. The technique involves proving a base case for the smallest possible value of the variable, and then assuming that the statement is true for some general value of the variable, and using that assumption to prove the statement for the next larger value of the variable. This process is continued until the statement has been proved for all values of the variable.