Day 03Part 04- Discrete mathematics – Short trick method for solving Nested Quantifiers problems.

Day 03Part 04- Discrete mathematics – Short trick method for solving Nested Quantifiers problems.

 Day 03 | Part 04 – Discrete Mathematics

Short Trick Methods for Solving Nested Quantifiers Problems

Nested quantifiers can be tricky, but with some simple techniques, you can solve them quickly and accurately. Let’s go through some short tricks and methods to solve nested quantifier problems in Discrete Mathematics.

 1. Understanding the Order of Quantifiers

The order of ∀ (Universal Quantifier) and ∃ (Existential Quantifier) matters in logic.

Rule 1:

∀x∃yP(x,y)≠∃y∀xP(x,y)∀x ∃y P(x, y) \neq ∃y ∀x P(x, y)∀x∃yP(x,y)=∃y∀xP(x,y)

 If Universal (∀) comes first, then for every x, we must find at least one y that satisfies P(x, y).
If Existential (∃) comes first, we must find at least one y that works for all x.

Shortcut:

• If the same type of quantifiers (∀…∀ or ∃…∃) appear together, order does NOT matter.
• If different quantifiers (∀, ∃) appear, order matters a lot!

 2. Quick Trick to Solve Nested Quantifier Problems

Trick 1: Consider a Simple Example (Use Small Values)

When solving a nested quantifier problem, try assigning small values to variables to check if the statement is true.

Example 1:

$$\forall x\exists y(x+y=10)$$

Step 1: Pick any x (say x = 3)
Step 2: Find some y such that $x+y=10$
Solution: If $x=3$, then $y=7$ works.
 Since we can always find such a y for every x, the statement is true.

Example 2:

$$\exists x\forall y(x+y=10)$$

Step 1: Find some x that works for all y.
Step 2: If $x=3$, then $3+y=10$ must be true for every y.
Impossible, because y can take any value, but $x+y$ should always be 10.
 This statement is false.

Trick 2: Swap Quantifiers and Check Meaning

If a question seems difficult, swap quantifiers and check how the meaning changes.

Example:

$$\forall x\exists yP(x,y)$$

 Means: “For every x, there exists some y such that P(x, y) is true.”
This is weaker than $\exists y\forall xP(x,y)$, because in the second case, we must find one single y that works for all x.

Trick 3: Use Counterexamples for False Statements

If you suspect a statement is false, try to find a counterexample.

Example:

$$\exists x\forall y(x-y>0)$$

 This means: “There exists some x such that for all y, $x-y$ is always positive.”
 But if we choose $x=5$, and y can be any number, what if $y=10$?
$5-10=-5$ (not positive) → This statement is false.

Conclusion: If you can find even one counterexample, the whole statement is false!

Trick 4: Convert Quantified Statements into English

Sometimes, converting statements into plain English makes them easier to understand.

Example:

$$\forall x\exists y(x\cdot y=1)$$

 Means: “For every x, there exists some y such that their product is 1.”
 If $x=0$, then $x\cdot y=1$ is impossible (because $0\times y=0$, not 1).
 So this statement is false.

 3. Quick Recap – Problem-Solving Strategy

Step 1: Identify order of quantifiers (Universal or Existential).
Step 2: Pick simple values to test if the statement holds.
Step 3: Try swapping quantifiers to see how the meaning changes.
Step 4: Look for counterexamples to disprove false statements.
Step 5: Convert the logical statement into plain English for better understanding.

 Conclusion

Nested quantifiers are essential for logical reasoning in computer science.
 Using small values, counterexamples, and translation tricks, we can solve problems faster.
Changing quantifier order significantly impacts the meaning of a logical statement.

 Do you want more practice problems or examples on this topic?

Day 03Part 04- Discrete mathematics – Short trick method for solving Nested Quantifiers problems.

Nested Quantifiers