Conditional Statement

Introduction

• Mathematical reasoning, a crucial skill that enables pupils to examine a given hypothesis without reference to a specific context or meaning, includes conditional statements.

• The reasoning is not based on an individual's opinion when a scientific investigation or assertion is investigated, to put it simply.

• The basis for deductions and proofs must be factual and scientific.

• To answer issues involving mathematical reasoning, one needs to possess critical thinking and logical reasoning abilities in mathematics.

• We will delve into the world of conditional statements in this quick session. Along with solved examples and interactive questions, we will go over the answers to queries like what is meant by a conditional statement, what are the pieces of a conditional statement, and how to make conditional statements.

Definition

• A conditional statement is one that takes the form "If p, then q." Here, "p" stands for "hypothesis," and "q" stands for "conclusion."

• For Example, If Lohit is hungry then, he has food.

• This sentence contains a condition. Another name for it is an implication. The relation between two statements is denoted by the symbol "→." For instance, $\mathrm{A\:\rightarrow\:B}$

• The logical connector is the name given to it. A and B can be regarded as being implied. Here are two additional instances of conditional statements.

Examples of Conditional Statements

• If a number is divisible by 9 then it is divisible by 3.

• If yesterday was Monday then today is Tuesday.

• If you are honest then you will not lie.

• If you are bold then you are a good performer

Truth Table

p	q	$\mathrm{p\:\rightarrow\:q}$
T	T	T
T	F	T
F	T	T
F	F	T

According to the conditional statement, if p is true, then q will logically follow, and be true as well. Therefore, the first row logically adheres to this definition. Similar to the first row, the second row comes after this because if the statement "p implies q" is true but q is false, then p is true, and the statement "p implies q" must be false because q didn't come right after p. P is false in row 3 but q is true.

Consider the following assertion. I put on my sunglasses when it's sunny. If p is false and q is true, then I donned my sunglasses despite the fact that it wasn't sunny. This in no way invalidates what I said in my first comment because I could just be a fan of sunglasses. Therefore, it makes sense to believe that "p implies q" is still true even if p is false but q is true. Row 4: P and q are false.

This would be the same as if it weren't sunny and I wasn't wearing my sunglasses from the previous example involving sunglasses. Once more, this wouldn't make my claim that "if it's sunny, I wear my sunglasses" incorrect. As a result, "p implies q" is still true even if p and q are both true.

Bi-Conditional Statements

• If and only if statements, which mathematicians like to abbreviate with "iff," are particularly potent since they essentially claim that the assertions p and q are interchangeable.

• When one is true, you might infer that the other is also true.

• In addition, if one is false, then the other must also be.

• The truth table reflects this. The biconditional is true whenever the truth values of the two statements coincide.

• If not, it is untrue. The double arrow in the biconditional really means "p implies q" and "q implies p." In a symbolic sense, it is the same as $\mathrm{p\:\longleftrightarrow\:q}$

Truth Table

p	q	$\mathrm{p\:\longleftrightarrow\:q}$
T	T	T
T	F	F
F	T	F
F	F	T

Solved Examples

Ray explains, "If a rectangle's perimeter is 12, then its area is 9." Which of the following could serve as the True statement? Defend your choice.

• A rectangle has sides that are 3 and 4 inches long.

• A square with sides that are 3 and 3.

• A rectangle having sides that are 2 and 6 in length.

• A rectangle with three and two-inch sides

Solution −

• Rectangle with sides 3 and 4: Perimeter = 14 and area = 12. Both 'if' and 'then' are false

• Rectangle with sides that are 3 and 3: Perimeter = 12 and area = 9. 'If' is true and 'then' is true.

• Rectangle with sides 2 and 6: Perimeter = 16 and area = 12. 'If' is false and 'then' is false.

• Rectangle with sides 3 and 2: Perimeter = 10 and area = 6 both if and then are false.

Write the converse, inverse, and contrapositive statements for the following conditional statement.

If you eat well, then you will be healthy.

Solution −

• The given statement is - If you eat well, then you will be healthy. It is of the form, "If p, then q".

• The converse statement is, "You will be healthy if you eat well" (if q, then p).

• The inverse statement is, "If you do not eat well then you will not be healthy" (if not p, then not q).

• The contrapositive statement is, "If you are not healthy, then you did not eat well" (if not q, then not p).

Conclusion

There are two categories of statements in the study of logic: conditional statements and bi-conditional statements. These assertions, which are referred to as compound statements, are created by combining two other statements. Consider the following statement: If it rains, we don't play. This is a synthesis of two assertions.

FAQs

1. What conditional statement is used the most frequently?

If is the conditional statement that is most frequently used. If statements should always be interpreted as "If X is TRUE, do a thing." By adding an otherwise clause, the logic is simply expanded to read "If X is TRUE, do this, or else do that."

2. What is the function of a conditional statement?

Conditional statements allow for the comparison of a hypothesis's premises and conclusions to ascertain whether the entire conditional statement is true.

3. What kind of operator does a conditional statement often use?

Similar to an if statement in that it evaluates boolean expressions, the conditional operator is a ternary operator (it has three operands), and if the test is true, it assigns a value to a variable rather than running a block of code.

4. What is the conditional statement's subject?

Conditional sentences explain factors that are known or hypothetical circumstances and their effects. Complete conditional sentences include both the result and the conditional phrase, also known as the "if-clause."

5. What is a conditional statement's other name?

Dependent, restricted, qualified, qualified, subject to, and with restrictions.