Definition Truth Table and Examples of If Then Statements in Proofs
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 Understanding If Then Statements in Mathematical Reasoning
1. What is an if-then statement in mathematical reasoning?
An if-then statement is a conditional statement that expresses a logical relationship in the form “If p, then q.” In mathematical reasoning, it connects a hypothesis to a conclusion.
- p is called the hypothesis (condition).
- q is called the conclusion (result).
- Symbolically written as p → q.
- Example: If a number is even, then it is divisible by 2.
2. What do hypothesis and conclusion mean in an if-then statement?
In an if-then statement, the hypothesis is the “if” part and the conclusion is the “then” part. The hypothesis states the condition, and the conclusion states what follows if that condition is true.
- Form: If p, then q.
- Hypothesis = p
- Conclusion = q
- Example: If x = 4, then x² = 16.
3. How do you write a conditional statement symbolically?
A conditional statement is written symbolically as p → q, which means “if p, then q.” This arrow represents logical implication in mathematical logic.
- p → q reads as “p implies q.”
- If p is true and q is false, the statement is false.
- In all other cases, the conditional is true.
4. Can you give an example of an if-then statement in mathematics?
An example of an if-then statement is: If a number is divisible by 4, then it is even. This shows a clear condition and result.
- Hypothesis: The number is divisible by 4.
- Conclusion: The number is even.
- Example check: 12 is divisible by 4, so 12 is even.
5. What is the converse of an if-then statement?
The converse of an if-then statement is formed by switching the hypothesis and conclusion. If the original statement is p → q, the converse is q → p.
- Original: If a shape is a square, then it has four sides.
- Converse: If a shape has four sides, then it is a square.
6. What is the contrapositive of a conditional statement?
The contrapositive of a conditional statement p → q is ¬q → ¬p, formed by negating and switching both parts. A statement and its contrapositive are logically equivalent.
- Original: If a number is divisible by 4, then it is even.
- Contrapositive: If a number is not even, then it is not divisible by 4.
7. What is the inverse of an if-then statement?
The inverse of a conditional statement p → q is ¬p → ¬q, formed by negating both the hypothesis and conclusion without switching them.
- Original: If it is a square, then it is a rectangle.
- Inverse: If it is not a square, then it is not a rectangle.
8. How do truth tables work for if-then statements?
A truth table shows all possible truth values of p and q to determine when p → q is true or false. The conditional statement is false only when p is true and q is false.
- p = T, q = T → T
- p = T, q = F → F
- p = F, q = T → T
- p = F, q = F → T
9. Why are if-then statements important in mathematical proofs?
If-then statements are important because most mathematical theorems are written in the form p → q. Proofs aim to logically show that whenever the hypothesis is true, the conclusion must also be true.
- Used in direct proofs.
- Used in proof by contrapositive.
- Used in proof by contradiction.
10. What are common mistakes when using if-then statements?
A common mistake is assuming that the converse of a true conditional statement is also true. Logical errors often occur when parts of the statement are incorrectly reversed or negated.
- Confusing converse with contrapositive.
- Assuming p → q means q → p.
- Forgetting that p → q is false only when p is true and q is false.
