 # Use of If Then Statements in Mathematical Reasoning

What is If and Then Statements?

Typically, a mathematical statement is made up of two compound components: the hypothesis aka assumptions, and the conclusion. These statements are actually two if and then statements Most mathematical statements you will come across in first year courses will have the form "If A, then B" or "A implies B". This suggests that the conditions that create "A" are the assumptions we form, while the conditions that create "B" are the conclusion. Thus, in order to prove that the –statement "If A, then B" is true, we would have to begin with making the assumptions "A" and then undertaking some tasks to ensure that conditions "A" are met so as to conclude "B".

Simply: If A and B implies positive integers

Then,

Product of A & B integers will also be positive

If - Then Statements

⤷Conditional Statements

If the Weather is Nice, then i will play Outside

↑                             ↑

Hypothesis              Conclusion

I get cake on my birthday.

Real Life Example of Using If and Then Statement

In everyday use, a statement devised as "If A, then B", often implies "A if and only if B." For example, when you ask your hiring manager "If you offer me a monthly salary of 40k, then I'll accept the job offer" they essentially it implies "I'll do your job if and only if you offer me 40k a month." Specifically, if you do not offer 40k/month, they won't be taking your job.

However, in mathematics, the statement "A implies B" is expressed in quite a different from ". These are often expressed in conditional statements of logical equivalence. Now, you must be wondering what is a conditional statement or what is the ‘if then statement’ in conditional statements.

Let’s quickly get to know about the ‘if and then’ conditional statements.

Identify Logically Equivalent Conditional Statements

If you findif and then” conditional statements to be challenging, we will help you elevate your LSAT skills significantly. Following our simple tricks and tips will help you easily determine the various equivalent ways that a true conditional statement can be demonstrated.

Let’s diagram conditional statements, so you find it more fun and fruitful finding logic.

Example 1

You are asked if your driver does not come to pick you, given the following two statements:

(i) If the driver does not come today then you will stay at home and complete your homework

(ii) You stayed at home and completed the homework

Using a diagram to express the conditional statement,

Now, Consider set theory that is just collections of objects illustrated by circles. If a set does not contain something, then it is not in the circle too. So,

If driver does not come today then you will stay at home and complete your homework

In which,

P = driver does not come today

Q = you stay at home and complete your homework

You stayed at home and completed the homework. This implies that ‘Q” is true. As per the diagram, if an object is inside ‘Q” it may or may not be inside. Thus, you can draw the inference nothing about the driver. Many people will want to inappropriately conclude that the driver must have not come, but conditional statements only move in one direction.

Truth Values of the Four Combinations

Here, we represent part by ‘P’ and then part by ‘Q’. Now, for the purpose of precisely explaining the truth value of a conditional statement, we require to take into account the four different combinations of the truth value for P and Q in relation.

1. ‘If P holds true, then Q too holds true. This statement is considered true given that if an object is inside a circle, then it is surely inside the circle.

2. If P holds true, then Q is false. This statement is considered false given that there is no viable way an object could be inside a circle and still outside the circle .

3. If P holds false, then Q clasp true. This statement is true as if an object is outside the circle then it may or may not be in the circle. There remains no strong contradiction.

4. If P holds false, then Q clasp false. This statement is also considered true because if an object is outside the circle, then it can be outside the circle. Like the previous statement, there is no contradiction.

## Truth Values of 4 Combinations Summarized in a Truth Table

 P Q P-Q T T T T F F F T T F F T

Solved Example

Problem

"Suppose that the bay is blazing in sunlight outside. Then there is a bright sun in the sky."

(i) Identify the assumptions and the conclusion.

(ii) Rewrite this statement clearly in the form "If A, then B" using Part (i).

(iii) What this statement holds— true or false?

Solution

(i) The hypothesis we are making is that it is shining bright outside, the conclusion we are making is that there must be golden sun in the sky.

(ii) "If it's the warmth of sunshine, then there must be a sun shining in the sky."

(iii) This statement is true. (Based on all that is presently known about how sunshine works! And how light and heat comes from the sun.

Did you know

A conditional statement is only false in case the ‘assumption’ is true and the ‘conclusion’ is false.

Any conditional statement with a false hypothesis is negligibly true.

1. What Kind of Reasoning Commonly Uses the IF THEN Statement?

The mathematical reasoning often uses if then statements. The reasoning is quite different from other reasoning like data interpretation or logical reasoning.  It relies on Deductive reasoning that often uses if then statements. A sentence is termed as a mathematically admissible statement if it is any of ‘true’ or ‘false’ but not both. By mathematical reasoning—the ‘if’ part is termed as hypothesis and the ‘then’ part is termed as conclusion. When deductive reasoning has been employed to prove an ‘if then statement’, then the fact that the hypothesis is true implies that the conclusion also holds true.

2. What is the Use of Truth Table?

Truth table is a mathematical table which is enormously workable for compiling complicated conditional statements and recognizing whether the statements are true or false. In a truth table, the contrapositive has the same truth value as the conditional till the cows come home. (Right! We mean always). This implies, if the conditional holds true then the contrapositive will too be true. However, the inverse does not hold true only because the conditional is true. The inverse will invariably have the same truth value as the converse.

Conditional Statement        If p, then q                         p → q

Converse                                If q, then p                         q → p

Inverse                                   If not p, then not q         ∼p → ∼q

Contrapositive                      If not q, then not p         ∼q →  ∼p

Biconditional                        p if and only if q                  p ↔️ q