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# Conclusion|Definition & Meaning

## Definition

The term conclusion in maths is used to define us about the problem that we solve and when we produce the final result at the end then that stage of processes is called as conclusion.

When you solve a **maths** question, you have to end the problem by calculating the last answer and pulling a **conclus****ion** by writing the answer. A conclusion is the **last step** of the maths **problem.** The conclusion is the** final **answer produced in the end**.** The **answer** is completed by writing the **arguments** and statements by telling the answer to the question. The ending statement of a problem is called a **conclusion.**

**Drawing conclusions** refers to the **act of thinking of interpreting a series of premises** or some ideas and, from them, suggesting something that leads to a meaningful finding. It is normally regarded as a **conscious way of learning**.

## Hypothesis and Conclusion

As a rule, a mathematical statement comprises **two sections**: the first section is **assumptions or hypotheses**, and the other section is the **conclusion**. Most mathematical statements have the form **“If A, then B.” **Often, this statement is written as **“A implies B”** or **“A **$\Rightarrow $ **B.”** The **assumptions** we make are what makes “A,” and the circumstances that make “B” are called the **conclusion**.

To prove that a given statement **“If A, then B”** is said to be true, we will require some **assumptions** for “A,” and after doing some work on it, we need to **conclude** that “B” must also hold when “A” holds.

If we are asked to apply the statement “If A, then B,” firstly, we should be **sure** that the conditions of the statement “A” are met and **true** before we start to talk about the **conclusion “B.”**

Suppose you want to apply the statement “x is even $\Rightarrow$ x2 is an integer.” First, you must **verify** that x is even **before** you **conclude** that x2 is an integer.

## If-then Statement

In maths, you will, at many times, confront statements in the form “X $\Leftrightarrow$ Y” or **“X if and only if Y.”** These statements are actually **two** “if, then” statements. The following statement, “X if and only if Y,” is logically **equivalent** to the statements “If X, then Y” and “If Y, then X.” One more method for thinking about this kind of explanation is an **equality** between the statements X and Y: so, whenever X holds, Y holds, and whenever Y hold, X holds.

Assume the example: “**x is even** $\Leftrightarrow$ **x2 is an integer**“. Statement **A** says, “x is even,” whereas statement **B** says, “x2 is an integer.” If we get a quick revision about what it suggests to be **even** (simply that x is a multiple of 2), we can see with ease that the following two statements are **identical**: If x=2k is proved to be even, then it implies x2=2k2=k is an integer, and we know that x2=k is an integer, then x=2k so n is **proved** to be even.

In day-to-day use, a statement which is in the form “**If A, then B**,” in some cases, means “**A if and only if B.**” For example, when people agree on a deal, they say, “If you agree to sell me your car for 500k, then I’ll buy from you this week” they straightaway mean, “I’ll buy your car if and only if you agree to sell me in 500k.” In other words, **if you don’t agree on 500k, they will not be buying your car from you**.

In geometry, the validation or proof is stated in the **if-then** format. The “if” is a **condition or hypothesis**, and if that condition is met, only then the second part of the statement is **true**, which is called the **conclusion**. The working is like any other **if-then statement.** For illustration, the statement “If a toy shop has toys for two age groups and 45 percent of toys in the shop are for 14 or above years old, then 55 percent of the toys in the shop are for 13 and fewer years old.” The above statement concludes that **“55 percent of the toys in the shop are for 13 and fewer years old.”**

## A implies B

In maths, the statement **“A if and only if B”** is ** very** different from

**“A implies B.”**Assume the example:

**“x is an integer”**is the A statement, and

**“x3 is a rational number”**is the B statement The statement “A implies B” here means “If x is an integer, then x3 is a rational number.” The statement is proven to be true. On the other hand, the statement, “A, if and only if B,” means “x is an integer if and only if x3 is a rational number,” which is not true in this case.

## Examples of Drawing Conclusions

### Example 1

Consider the equation below. Comment if this equation is true or false.

### Solution

To calculate its true answer, first, consider the hypothesis $x>0$. Whatever we are going to conclude, it will be a consequence of the truth that $x$ is positive.

Next, consider the conclusion $x+1>0$. This equation is right, since $x+1>x>0$.

This implies that the provided inequality is true.

### Example 2

Simplify the below problem by providing a **conclusion** by calculating the **answer** of A.

\[ A= \dfrac{35}{3} \]

### Solution

The **expression** given in the **question** is: $A= \dfrac{35}{3}$

**Calculating** the answer of A to make a **conclusion,** The arithmetic operation **division** is found in the question that is to be **figured** out in the provided problem. After figuring out the answer to **expression** A, The **conclusion** will be given.

\[ A= \dfrac{35}{3} \]

\[ A= 11. 667 \]

**Therefore,** we **conclude** the question by calculating the **answer** of $A=11.666$

### Example 3

Consider the equation $0>1 \Rightarrow sinx=2$. Is this equation true or false?

### Solution

To calculate the correct answer, first consider the hypothesis $0>1$. This equation is clearly false.

### Example 4

calculate the below problem by providing a conclusion by estimating the value of X.

\[ 3+8 \times 2\]

### Solution

The expression given in the problem is $3+8 \times 2 $.

Multiplication and Plus operation is to be carried out to calculate the answer to the given problem. After figuring out the answer to X the conclusion will be given.

\[ 3+8 \times 2\]

\[ 3+ 16\]

\[ 19\]

Thus, we conclude the example by calculating the value of $X = 19$.

*All images/mathematical drawings were created with GeoGebra.*