How If Then Statements Build Strong Mathematical Arguments
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 find “if 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,
[Image will be Uploaded Soon]
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.
‘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.
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 .
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.
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
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.
FAQs on If Then Statements in Mathematical Reasoning: A Complete Guide
1. What is an 'if-then' statement in the context of mathematical reasoning?
An 'if-then' statement, formally known as a conditional statement, is a fundamental logical structure used in mathematical reasoning. It connects two smaller statements, a hypothesis and a conclusion, in the form 'If p, then q'. The statement asserts that the conclusion 'q' must be true whenever the hypothesis 'p' is true. According to the rules of logic, this entire statement is only considered false when the hypothesis is true, but the conclusion is false.
2. What are the two primary components of an 'if-then' statement?
Every 'if-then' statement, which can be symbolised as p → q, is composed of two essential parts:
Hypothesis: This is the 'if' part of the statement, represented by 'p'. It sets up the condition or premise for the argument.
Conclusion: This is the 'then' part of the statement, represented by 'q'. It is the result or outcome that is claimed to follow from the hypothesis.
For instance, in the statement, 'If a triangle has two equal sides, then it is an isosceles triangle,' the hypothesis is 'a triangle has two equal sides,' and the conclusion is 'it is an isosceles triangle.'
3. Can you provide some examples of 'if-then' statements used in mathematics?
Certainly. 'If-then' statements are crucial for expressing mathematical theorems and properties. Here are a few examples:
Algebra: If x = 5, then 2x - 3 = 7.
Geometry: If a polygon is a square, then it has four right angles.
Number Theory: If an integer is divisible by 6, then it is divisible by 3.
4. What is the difference between the converse, inverse, and contrapositive of an 'if-then' statement?
For any given conditional statement 'If p, then q', we can derive three related logical statements. Understanding their differences is key to mathematical reasoning:
The Converse swaps the hypothesis and conclusion: 'If q, then p'.
The Inverse negates both parts: 'If not p, then not q'.
The Contrapositive negates both parts and then swaps them: 'If not q, then not p'.
A statement and its contrapositive are always logically equivalent, meaning they are either both true or both false. However, a statement is not logically equivalent to its converse or inverse.
5. How does a truth table help in analysing 'if-then' statements?
A truth table is a vital tool for systematically evaluating the validity of a conditional statement. It displays the truth value (True or False) of the statement 'If p, then q' for all possible combinations of truth values of p and q. The key takeaway from the truth table is that a conditional statement is only false in one scenario: when the hypothesis (p) is true and the conclusion (q) is false. In all other cases, the statement is considered true. This helps in verifying logical equivalences and validating arguments.
6. Why is it a logical error to assume the converse of a true 'if-then' statement is also true?
Assuming the converse is true is a common logical fallacy because the truth of a conditional statement (p → q) does not guarantee the truth of its converse (q → p). For example, the statement 'If a shape is a square, then it is a rectangle' is true. However, its converse, 'If a shape is a rectangle, then it is a square,' is false. Many rectangles are not squares. In mathematical proofs, it is crucial to prove the conditional and its converse separately if you need to establish a biconditional relationship ('if and only if').
7. How does an 'if-then' statement relate to a biconditional statement?
A biconditional statement, written as 'p if and only if q' (or p ↔ q), is a stronger form of logical connection. It essentially combines a conditional statement with its converse. For a biconditional statement to be true, both the original 'if-then' statement ('If p, then q') and its converse ('If q, then p') must be true. It signifies that p and q are logically equivalent and always have the same truth value.
