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Interactive Audio Lesson
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Disproving Universal Quantified Statements
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Today, we will learn about how to disprove universally quantified statements. Can anyone tell me what a universally quantified statement is?
Isn't it a statement that asserts something is true for all elements in a domain?
Great! Exactly. So, to disprove such a statement, we need a counterexample, which is an instance where the statement does not hold. Can anyone think of an example?
What about the statement that every positive integer is a sum of two squares? We can use 3 as a counterexample since it's not expressible that way.
Precisely! 3 is a counterexample. Remember, if P(x) is false for just one x, then orall x P(x) is also false. That’s a key take-home lesson!
Let's recap: what do we need to disprove a universal statement?
One counterexample!
Proof by Cases
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Let's move on to proof by cases. Who can explain what this entails?
It's where we examine all possible scenarios to prove a statement!
Exactly! If you want to prove p → q, and P can be divided into n cases (P₁, P₂,..., Pₙ), we must show that q follows from each sub-case. Can anyone provide an example where this would apply?
What about proving n² ≥ n for all integers n? We can look at negatives, zero, and positives as different cases.
Precisely! For each case: if n = 0, positive, or negative, we verify the inequality holds true. This method is effective in discrete mathematics!
Now, can anyone summarize why proof by cases is useful?
Because we cover all possibilities and ensure our proof is robust!
Without Loss of Generality (WLOG)
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Now, let’s discuss without loss of generality, or WLOG. What does that mean?
It means we can assume one scenario is true without losing the generality of the proof!
Perfect! For instance, when proving properties about integers x and y, if we show it for x ≥ y, it covers both situations due to symmetry. Can you give a scenario where WLOG simplifies our steps?
If proving x + y is even, we could just choose to show it for x being even without needing to explore the case where y is even.
Exactly! WLOG allows us to eliminate redundancy in proofs. Let’s summarize.
By assuming certain conditions hold true, we simplify proofs without losing arguments’ integrity. What have we learned?
Assuming one scenario can save time and effort while proving some properties!
Constructive and Non-Constructive Proofs
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Now let’s explore constructive and non-constructive proofs. What’s the key distinction?
A constructive proof provides a specific example showing existence, while a non-constructive proof doesn’t give examples but proves existence logically.
I see! In this case, we can’t provide examples but still prove their existence.
Exactly right! Remember, in non-constructive proofs, we’re still showing definitive logic without an explicit witness. Can anyone summarize the main points?
Constructive gives specific examples, and non-constructive proves logically without examples.
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
In this section, we deepen our understanding of proof strategies, focusing on how to disprove universally quantified statements through counterexamples, employing proof by cases, and distinguishing between constructive and non-constructive proofs for existential statements.
Detailed
Proof Strategies-II
In the continuation of proof strategies, this lecture delves into essential methods used in discrete mathematics for validating or invalidating statements. Fundamental topics include:
Disproving Universal Quantified Statements
- To disprove a universally quantified statement like "All P are Q", one must provide a counterexample from the domain where the statement does not hold true. For instance, the statement "Every positive integer can be expressed as a sum of squares of two integers" can be falsified by proving that the integer 3 cannot be expressed in that form. This approach emphasizes the critical distinction between proving and disproving through examples.
Proof by Cases
- A strategy known as proof by cases is outlined, designed to validate implications of the form "if P, then Q" (p → q), where P can be expressed as multiple parts (P₁, P₂, ..., Pₙ). An example demonstrated involves proving that for any integer n, n² ≥ n, by separately analyzing cases when n is negative, zero, and positive. This method is particularly useful for situations internal to various discrete domains.
Without Loss of Generality (WLOG)
- The term without loss of generality (w.l.o.g.) is introduced, aiding in reducing complex proofs by allowing the examination of one representative case without affecting the integrity of universality in results.
Proving Existential Quantified Statements
- Existential quantified statements can be proven in two ways:
- Constructive Proof: Where a specific example, like the number 1729, is provided to show its properties.
- Non-Constructive Proof: Where one logically argues the existence of a solution without providing an explicit example, illustrated through irrational numbers.
Uniqueness Proofs
- Uniqueness proofs necessitate demonstrating both existence of a solution and establishing that no alternative solutions exist.
Backward Reasoning
- A novel proof strategy known as backward reasoning enables the proof of a statement by establishing a true antecedent p from which the conclusion q can flow. This reverse verification method solidifies understanding of ranged statements.
In concluding the lecture, we explored these diverse mechanisms that will be encountered throughout the course, each playing a vital role in assessing the validity of mathematical statements.
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Disproving Universally Quantified Statements
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In this lecture we will continue our discussion on the proof strategies that we started in the previous lecture.
We will start with how to disprove a universally quantified statement. So till now we were seeing various proof methods, where we wanted to prove universally quantified statements. But sometimes we also encounter statements where we want to prove that a universal statement, universally quantified statement, is not true. To disprove that a universally quantified statement is not true, we have to give what we call as a counterexample.
And, by counterexample we mean some element c from the domain such that the statement P is not true for the element c, because if the statement P is not true for the element c. Then the universal quantification for all x, P(x) will turn out to be false. So, very simple example of this is if I make a statement that every positive integer can be expressed as the sum of squares of 2 integers. So this property is my property P and I am making this statement for every x. So, now I have to verify whether the statement P is true for every integer x or not. Even if I can find one integer for which this property is not true, I can say that ꓯ x, P(x) is false here and a very simple counterexample is 3 here, you can check that 3 can never be expressed as a sum of squares of 2 integers.
Detailed Explanation
To disprove a statement that claims something is true for all elements of a certain set, you can show that there is at least one exception. This is done by providing a counterexample, which is a specific instance where the statement fails to hold. In the example given, the statement claims every positive integer can be represented as the sum of two squares. By finding the integer 3, which cannot be expressed as such, we can conclusively say that the original statement is false.
Examples & Analogies
Think of a claim like 'All birds can fly.' If you find just one bird that cannot fly, like a penguin, you have disproven the claim. Similarly, if someone claims that all students in a class passed the test, but you find one who failed, the claim becomes false. This shows how counterexamples work in everyday reasoning.
Proof by Cases
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However, I would like to stress here that there is nothing called proof by example for proving universally quantified statements. If you want to prove that your statement P is true for every element in the domain, then you cannot say that okay, I am showing it for some x, where x is explicitly chosen by you, it is not arbitrary. You are not choosing your x arbitrary, you are just taking your x specifically and that is an example for you.
There is another proof mechanism which is called proof by cases. This is also called exhaustive proof. So if you want to prove an implication of the form p → q, where p and q are propositions, and if your proposition P can be decomposed into conjunction of various other propositions say n propositions, then p → q is logically equivalent to p → q, conjunction 1 p → q and so on.
So if you take the case n equal to 0 the statement is true, namely p → q is true. If you take the case when your arbitrarily chosen n is positive, then also this statement p → q is true and if your arbitrarily chosen n is negative, then you show that in that case also this p → q is true and since these are the three possible cases for your arbitrarily chosen n you can conclude that p → q is true.
Detailed Explanation
Proof by cases involves dividing a problem into several distinct scenarios (or 'cases') and proving that the statement holds true for each scenario. This is particularly useful when you are dealing with statements that can be true under different conditions. For example, if you want to prove a mathematical statement for all integers, you could consider the cases where the integer is positive, negative, or zero. If the statement is confirmed true in each case, it is true for all integers.
Examples & Analogies
Imagine you're planning a party and need to know how many snacks to get depending on whether there are kids, adults, or a mix of both attending. You could make a plan for each case: one plan for ages 0-12, another for ages 13-18, and a third for those 19 and older. If your snack plan works for each scenario, you can be confident it will be effective for the entire party. This is akin to proof by cases in mathematics.
Without Loss of Generality
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There is also something called without loss of generality, which we very often encounter while writing the proof. In short form they call it as w.l.o.g. So let me demonstrate we have what exactly I mean by without loss of generality. So imagine I want to prove that if x and y are integers and both x, y and x + y are even, then x and y are both even.
Detailed Explanation
Without loss of generality (w.l.o.g.) is used in proofs to simplify arguments by assuming a specific case that doesn’t limit the generality of what you are proving. For example, if you are proving a statement about two integers x and y, you can assume x is less than or equal to y without loss of generality. This means that you can simplify your proof without affecting the overall conclusion.
Examples & Analogies
Imagine you are studying T-shirt sizes in a store—small, medium, and large. If you wanted to prove that everyone should try at least one size, you could just show that the small works. With w.l.o.g., you'd argue that if the small fits, so will the medium and large since they’re only larger. By proving that one size works, you’ve effectively demonstrated the point for all sizes without needing to try on each one.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Disproving Universal Statements: Using counterexamples to show a universal statement is false.
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Proof by Cases: Breaking down a proof into distinct cases where each is shown true.
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Without Loss of Generality: Assuming one case suffices for all similar cases.
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Constructive Proof: Providing specific instances that demonstrate existence.
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Non-Constructive Proof: Proving existence without specific examples.
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Uniqueness Proof: Proving an element exists and is the only one satisfying its property.
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Backward Reasoning: Demonstrating a conclusion by starting with an established premise.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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The statement that all positive integers can be expressed as a sum of two squares is disproved by the counterexample 3.
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Using proof by cases, we show that n² ≥ n for all integers by checking values for n < 0, n = 0, and n > 0.
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In proving x + y is even, we can use WLOG by stating if x is even, y is odd suffices for showing the result.
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To demonstrate that there exists a number that can be expressed in two ways as the cube of sums, we can state 1729.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
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Counterexample proves, the falsely moves; cases split the proof like shoes in twos.
📖 Fascinating Stories
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In a village, there lived a wise man who proved every fact. To validate 'All are happy,' he sought a sad soul named Ann—as the only exception made his proof untrue!
🧠 Other Memory Gems
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WLOG: Where One Looks Often Gains simplicity in proof!
🎯 Super Acronyms
CPC
- Counterexamples
- Proofs by cases
- and Constructive proofs are key for successful proofs!
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
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Term: Counterexample
Definition:
An example that disproves a statement or proposition.
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Term: Proof by Cases
Definition:
A method of proving statements by checking all possible cases.
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Term: Without Loss of Generality (WLOG)
Definition:
A technique that allows one to assume a simplified case without affecting the general validity of the argument.
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Term: Constructive Proof
Definition:
A proof that provides a specific example of a case that satisfies the condition.
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Term: NonConstructive Proof
Definition:
A proof that asserts existence without providing a specific example.
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Term: Unicity Proof
Definition:
A method of showing that only one element satisfies a property.
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Term: Backward Reasoning
Definition:
A strategy of proving a statement by establishing the truth of a premise that leads to it.