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# ICT 101 Use of Logical Equivalence and Its Features Assessment 4 Answer

**ICT**** ****101**** ****Discrete**** ****Mathematics**** ****for**** ****IT**

**Assessment # 4 - Group Assignment**** ****Trimester**** ****3,**** ****2019**

**Overview**** ****of**** ****Assignment**

This assignment will test the students’ skill on writing literature review of a topic you have learnt in the Discrete Mathematics (ICT101) course in the week 1 to 6. Student need to read at least 3 articles or books on this topic especially with application to Information Technology and give detail review of those. Student will also identify one particular application of information Technology related to the topic in which he/she is interested and write a complete account of that interest.

**Task**** ****Description:**** ****Written**** ****Report**

Student will be exploring one Mathematical Topic and its uses in Information Technology. Students are to write a report on their findings.

Student can choose one from the following Topic. However, after deciding on the topic to work on, student must meet his/her respective tutor to know which topic was chosen and to get some guidance.

**The**** ****problems your**** ****group**** ****can**** ****choose**** ****from**** ****are:**

- Arithmetic operations in Binary Number System
- Logical Equivalence
- Proof technique
- Inverse function
- Linear Recurrences
- BCD Arithmetic

**The**** ****written**** ****report**** ****must**** ****have**** ****the**** ****following**** ****sections:**

- Introduction
- Proper reference of at least three articles or books
- Write detail review of those articles or books related to the topic student chooses
- Identify one particular application in Information Technology in which student is interested. Write a complete account of that interest
- Conclusion

## Answer

**Introduction**

In this assignment, we select the topic Logical equivalence. In Logical equivalence two mathematical statements or logics are compared. These statements are compared under some set of formulas and checked for their truth values in a model. The equivalency is then represented by (:: or double arrow) sign. Equivalency are of many types and uses and different from each other but their basic concepts are same.

**Logical equivalence**

Logical equivalency is all about to compare two mathematical statements and said that they are equivalent or not based their truth values. It is used to check the equality of a set of mathematical values and compare their relation. It compares their truth values, if the value is same then the statements are logically equivalent and if the truth values are different, the statements are un-equal. This logical equivalence comparison, is used in providing mathematical results by replacing a mathematical expression with alternative equal expression, in this situation, the truth value of the original unit, will not change. We use three types of propositions based on their truth values that are Tautology, contingency and contradiction. A proposition that is anytime a resulted a true value is called Tautology. A proposition that is anytime resulted a false value is called Contradiction and a proposition that will anything from a tautology or a contradiction is called a Contingency

To understand it with details, we can take an example. There are two propositions namely P and Q. These are said to be logical equivalent. This will be shown as P ↔ Q and resulted as a Tautology statement. The mostly used method of proving that two propositions are logically equivalent by using truth tables. The used truth table is same for all the comparing values used in that set. In some cases, this method of using truth table is not practicable. Because there will be different set of complexities in the variables of propositional, we are comparing in the terms of used size or expression values. Beside using the above method, there are other better ways to prove that the two given values are logical equivalent. In this method, we construct a mathematical proof that is using pre derived logical equivalence values. These values will result more specific and useful logical equivalence values. Here are some bi conditional and conditional values that are used in this method.

From the above understanding of methods and formulas used to derive logical equivalence propositions, we can take an example to solve a mathematical formula from some pre defined laws. An equation is provided to us that need to be prove for logical equivalency using mathematical method. The provided equation is – **¬ (P → Q) ≡ P ^ ¬ Q**. to solve this equation, we consider its left hand side values that is –

**¬ (P → Q) ≡ ¬ (¬ P ^ Q)** ………………………………………… taken from first equivalence condition

** ≡ ¬ (¬ P) ^ ¬ Q** …………………………………….. derived from De Morgan’s law

** ≡ P ^ ¬ Q** ……………………………………………. Derived from Double negation law

So that, it is proved that the equation id logically equivalent by its left hand side to right hand side values.

**Information technology application**

There are so many mathematical applications today that are used in the information technology that can ease a complex task. The computer algebra system is one of the most used application or we can say a package of software that is used to resolve maths problems. This application will automatically calculate a difficult algebraic task and solve it within a second. It is slightly different from the traditional calculator by its additional functionality of using numeric values with symbols. Its capabilities may be varied from system to system but the basic functionalities will be remained same. This application has features of graphically design an equation. Its interface delivers a programming language that helps one in designing own tasks as a design and used. This software application is today used by many institutes ad universities that changes their way of teaching mathematics and started using this flexible tool to solve mathematical problems. This application is used to solve functions of rational, polynomials factors, solution of an equation and so many other maths issues. To resolve these type of math problems, it has in built tools like Maple, MathCAD or Mathematica. For example, when we try to solve a simple math problem that is so long to calculate with the help of a calculator and finally shown an error. This daunting task can be easily solved by the help of Maple tool within few seconds and with accuracy in result. We can also track the steps used in resolving a math problem. This software application eliminates the involvement of humans by physically in solving typical mathematical problem and every thing will be manage and derived by using tools and software.

**Conclusion**

In this assignment, we will discuss the use of logical equivalence and its features. Logical equivalency is sued to compare two mathematical values and check or compare their relation by their truth values. We use three types of propositions based on their truth values that are namely Tautology, contingency and contradiction. There are several methods presents to check the two values are logically equivalent or not. These methods include truth tables, mathematical tables etc. By using these tables, we can easily calculate and derive result of a complex math problem. We preferred to use a mathematical application that is mostly used in information technology today to resolve easy to complex maths problems, is computer algebra system. This software application has so many tools to help resolve maths issues within a second. Today, so many institutes and universities are using this application in their learning and making easy the complex mathematical problems.

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