Mathematics: Predicates and Quantifiers

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Questions and Answers

What characterizes an ordered list compared to a sorted list?

An ordered list may have the same values but is arranged in a specific sequence. (correct)

Both lists contain the same values in the same order.

An ordered list must have unique values.

A sorted list has more elements than an ordered list.

For the sequence S = [4, 13, 72, 89, 51], what does the predicate ∃𝑖 ∈ {1, … , 5} ∙ 𝑆(𝑖) = 89 indicate?

There are multiple indices where the value 89 appears.

89 is the first term of the sequence.

89 is present in the sequence but its position is unknown. (correct)

The sequence ends with the number 89.

Which sequence is described as sorted in ascending order?

S = [5, 21, 21, 52, 52]

S = [4, 17, 28, 32, 90] (correct)

S = [15, 41, 89]

S = [90, 32, 28, 17, 4]

Which statement accurately describes finite sequences?

Finite sequences have a predetermined number of terms. Signup and view all the answers

What does the notation ∀𝑖 ∈ {1, … , 4} ∙ 𝑆(𝑖) < 𝑆(𝑖 + 1) represent?

The sequence is sorted in ascending order. Signup and view all the answers

In the context of sequences, what does the term 'index' refer to?

The position of a term within the sequence. Signup and view all the answers

Which of the following is true about an infinite sequence?

It allows for the possibility to predict future terms based on a pattern. Signup and view all the answers

What is the primary difference between the sequences [15, 41, 89] and [41, 89, 15]?

Both sequences contain exactly the same values but are in different orders. Signup and view all the answers

What does the statement ∃𝑥∈𝐷·𝑤𝑜𝑚𝑎𝑛 𝑥 ⇒ 𝑡𝑎𝑙𝑙(𝑥) imply?

There exists something that might not be a woman, but is tall. Signup and view all the answers

How would you interpret the predicate ∃𝑥∈𝐴·∃𝑦∈𝐵·𝑥 = 𝑦?

There exists at least one x in A and one y in B such that x equals y. Signup and view all the answers

What does the statement ∀𝑥∈𝐴·∃𝑦∈𝐵·𝑥 < 𝑦 mean?

For every x in A, there exists a y in B such that x is less than y. Signup and view all the answers

What is a consequence of the statement ∃𝑥∈𝐷·𝑤𝑜𝑚𝑎𝑛 𝑥 ∧ 𝑡𝑎𝑙𝑙(𝑥)?

There exists at least one individual who is both a woman and tall. Signup and view all the answers

What is the implication of using the symbol 'L(a, b)', where L stands for love?

A loves B and B loves A. Signup and view all the answers

What logical notation represents 'There exists an even integer'?

∃𝑥∈ℤ: 𝐸(𝑥) Signup and view all the answers

How would you express 'Every integer is even or odd' in logical notation?

∀𝑥∈ℤ: 𝐸(𝑥) ∨ 𝑂(𝑥) Signup and view all the answers

What does the notation ∀𝑥∈ℤ: 𝐸(𝑥) ∧ 𝑃(𝑥) ⇒ 𝑥 = 2 represent?

There is only one even prime Signup and view all the answers

Which of the following represents 'Not all integers are odd'?

∃𝑥∈ℤ: ¬𝑂(𝑥) Signup and view all the answers

How can 'Not all primes are odd' be expressed logically?

∃𝑥∈ℤ: P(𝑥) ∧ ¬𝑂(𝑥) Signup and view all the answers

Which logical statement correctly states that all prime integers are non-negative?

∀𝑥∈ℤ: P(𝑥) ⇒ N(𝑥) Signup and view all the answers

What does the statement 'If an integer is not odd, then it must be even' represent in logical notation?

∀𝑥∈ℤ: ¬O(𝑥) ⇒ E(𝑥) Signup and view all the answers

Which notation shows that not every person in the class csc1026 gets the train?

∃𝑥∈𝐷: ¬Q(𝑥) Signup and view all the answers

What is the correct representation of 'Everyone in csc1026 has lectures on Thursday and Friday'?

∀𝑥∈𝐷: P(𝑥) Signup and view all the answers

What does the logical statement ∃𝑥∈ℤ: 𝑃(𝑥) ∧ ¬𝑂(𝑥) convey?

Some primes are even Signup and view all the answers

What condition does the predicate ∀𝑖 ∈ {1, … , 4} ∙ (𝑆(𝑖) ≤ 𝑆(𝑖 + 1)) ∨ (𝑆(𝑖) ≥ 𝑆(𝑖 + 1)) express for a sequence?

The sequence is sorted in either ascending or descending order. Signup and view all the answers

Which example satisfies the predicate ∀𝑖 ∈ {1, … , 4} ∙ (𝑆(𝑖) ≤ 𝑆(𝑖 + 1)) ∨ (𝑆(𝑖) ≥ 𝑆(𝑖 + 1)) but is not fully sorted?

[6, 2, 91, 34, 62] Signup and view all the answers

In the context of the sequence S = [14, 5, 71, 22, 7], what does the predicate ∃𝑖 ∈ 1, … , 5 ∙ ∀𝑗 ∈ {1, … , 5} ∙ 𝑆(𝑖) ≥ 𝑆(𝑗) imply?

There exists an index whose corresponding value is the greatest in the sequence. Signup and view all the answers

What does the predicate ∀𝑖 ∈ ℕ ∖ {0} ∙ 𝑆(𝑖) = 2 ∗ 𝑖 + 1 represent?

The sequence of all odd integers greater than or equal to 1. Signup and view all the answers

Which correctly describes the Fibonacci sequence?

Each term is the sum of its two preceding terms. Signup and view all the answers

Why does the predicate ∀𝑖 ∈ {1, … , 4} ∙ (𝑆(𝑖) ≤ 𝑆(𝑖 + 1)) ∨ (∀𝑗 ∈ {1, … , 4} ∙ 𝑆(𝑗) ≥ 𝑆(𝑗 + 1)) need to be modified?

It does not account for combinations of increasing and decreasing values. Signup and view all the answers

To determine potential subsequent values in an observed pattern of an infinite sequence, what is essential?

There must be an identifiable rule that governs the pattern. Signup and view all the answers

Which of the following is true about infinite sequences discussed?

They can often be predicted by identifying patterns. Signup and view all the answers

What does the expression ∀𝑥∈𝐷∃𝑦∈𝐷𝐿(𝑦,𝑥) signify?

Everyone is loved by someone. Signup and view all the answers

What is the negation of the statement 'All dogs have fleas' expressed in logical terms?

∃𝑥∈𝑑𝑜𝑔𝑠 ¬𝑓𝑙𝑒𝑎𝑠(𝑥) Signup and view all the answers

Which expression means 'Someone loves someone' in logical notation?

∃𝑥∈𝐷∃𝑦∈𝐷𝐿(𝑥,𝑦) Signup and view all the answers

What is the logical implication of the statement '∀𝑥∈𝐷𝐿(𝑥,𝑥)'?

Everyone loves themselves. Signup and view all the answers

Which statement represents 'Someone is loved by everyone' logically?

∃𝑥∈𝐷∀𝑦∈𝐷𝐿(𝑦,𝑥) Signup and view all the answers

What is the correct negation of the statement 'Some drivers do not obey the speed limit'?

∀𝑥∈𝑑𝑟𝑖𝑣𝑒𝑟𝑠𝑜𝑏𝑒𝑦𝐿𝑖𝑚𝑖𝑡(𝑥) Signup and view all the answers

Which logical expression represents 'Everyone loves everyone'?

∀𝑥∈𝐷∀𝑦∈𝐷𝐿(𝑥,𝑦) Signup and view all the answers

What is the outcome of applying De Morgan's rule to ¬∃𝑥∈𝐷∀𝑦∈𝐷∀𝑧∈𝐷𝑃(𝑥,𝑦,𝑧)?

∀𝑥∈𝐷∃𝑦∈𝐷¬∀𝑧∈𝐷𝑃(𝑥,𝑦,𝑧) Signup and view all the answers

Study Notes

Interpreting Predicates

To prove the existence of an element, just give one example
The English statement "There is an integer whose square is twice itself" can be expressed in logic as ∃𝑥𝑥 2 = (𝑥 ∗ 2)
The integer that satisfies the statement is 0

Quantifiers

∀𝑥𝐷𝑃 𝑥 is the same as: 𝑃 1 ∧ 𝑃(2) ∧ 𝑃(3) when D = { 1, 2, 3 }
∃𝑥𝐷𝑃 𝑥 is the same as: 𝑃 1 ∨ 𝑃(2) ∨ 𝑃(3) when D = { 1, 2, 3 }

Exercises

The Domain is Z, the set of Integers
N(x) : x is a non-negative integer
E(x) : x is even
O(x) : x is odd
P(x) : x is a prime number
There exists an even integer: ∃𝑥 𝐸(𝑥)
Every integer is even or odd: ∀𝑥 𝐸(𝑥) ∨ 𝑂(𝑥)
All prime integers are non-negative: ∀𝑥 𝑃 𝑥 ⇒ 𝑁(𝑥)
The only even prime is 2: ∀𝑥 𝐸 𝑥 ∧ 𝑃 𝑥 ⇒ 𝑥 = 2
Not all integers are odd: ∃𝑥 ¬𝑂(𝑥)
Not all primes are odd: ∃𝑥 𝑃 𝑥 ∧ ¬𝑂(𝑥)
If an integer is not odd, then it must be even: ∀𝑥 ¬𝑂 𝑥 ⇒ 𝐸(𝑥)

Clarification from Earlier

∀𝑥𝐷𝑥 𝜖 𝑐𝑠𝑐1026 ⇒ 𝑃 𝑥 - Everyone in csc1026 has lectures on Thurs & Fri
∃𝑥𝐷𝑥 𝜖 𝑐𝑠𝑐1026 ∧ 𝑄(𝑥) - Some people in csc1026 get the train to come to university
∃𝑥𝐷 𝑤𝑜𝑚𝑎𝑛 𝑥 ∧ 𝑡𝑎𝑙𝑙(𝑥) - There exists a tall woman

Combining Quantifiers

Let A = {2, 6, 14} and B = {6, 8, 11}
∃𝑥𝐴 ∃𝑦𝐵 𝑥 = 𝑦 - This means: "There exists an x in A such that there exists a y in B such that x = y."
∀𝑥𝐴 ∃𝑦𝐵 𝑥 < 𝑦 - This means: "For every x in A, there exists a y in B such that x < y."

The Lovers Relationship

L(a,b) be “a loves b”
∀𝑥𝐷 ∃𝑦𝐷 𝐿 𝑦, 𝑥 - Everyone is loved by someone
∀𝑥𝐷 ∃𝑦𝐷 𝐿 𝑥, 𝑦 - Everyone loves someone
∃𝑥𝐷 ∀𝑦𝐷 𝐿 𝑥, 𝑦 - Someone loves everyone
∃𝑥𝐷 ∀𝑦𝐷 𝐿 𝑦, 𝑥 - Someone is loved by everyone
∃𝑥𝐷 𝐿 𝑥, 𝑥 - Someone loves them self
∀𝑥𝐷 𝐿 𝑥, 𝑥 - Everyone loves themselves
∃𝑥𝐷 ∃𝑦𝐷 𝐿 𝑥, 𝑦 - Someone loves someone
∃𝑥𝐷 ∃𝑦𝐷 𝐿 𝑦, 𝑥 - Someone is loved by someone
∀𝑥𝐷 ∀𝑦𝐷 𝐿 𝑥, 𝑦 - Everyone loves everyone
∀𝑥𝐷 ∀𝑦𝐷 𝐿 𝑦, 𝑥 - Everyone is loved by everyone

Exercise

"Some drivers do not obey the speed limit" - ∃𝑥𝑑𝑟𝑖𝑣𝑒𝑟𝑠 ¬𝑜𝑏𝑒𝑦𝐿𝑖𝑚𝑖𝑡(𝑥)
"All dogs have fleas" - ∀𝑥𝑑𝑜𝑔𝑠 𝑓𝑙𝑒𝑎𝑠 𝑥

Negation of Multiple Quantifiers

Basic rule for negation of quantified predicates: Flip the quantifier and negate the predicate
Example: ¬∃𝑥𝐷∀𝑦𝐷∀𝑧𝐷𝑃(𝑥, 𝑦, 𝑧) = ∀𝑥𝐷 ∃𝑦𝐷 ∃𝑧𝐷 ¬𝑃(𝑥, 𝑦, 𝑧)

Sets vs Sequences

A set is an unordered collection of objects
A sequence is an ordered list of objects

Ordered?

A sequence is an ordered list of objects, not necessarily sorted
In the sequence S = [15, 41, 89], S(1) = 15
In the sequence T = [41, 89, 15], T(1) = 41

Length of a Sequence

A sequence could be finite or infinite

Predicates over Sequences

For a sequence of finite length, limit the domain to the number of terms in the sequence
Given the sequence S=[4, 13, 72, 89, 51], ∃𝑖 ∈ {1, … , 5} ∙ 𝑆(𝑖) = 89 states that 89 is in the sequence, but not where

Sorted

A sorted sequence is arranged in ascending or descending order
For a 5 term sequence: ∀𝑖 ∈ {1, … , 4} ∙ 𝑆(𝑖) ≤ 𝑆(𝑖 + 1)
For a sequence sorted either ascending or descending: (∀𝑖 ∈ {1, … , 4} ∙ 𝑆(𝑖) ≤ 𝑆(𝑖 + 1)) ∨ (∀𝑗 ∈ {1, … , 4} ∙ 𝑆(𝑗) ≥ 𝑆(𝑗 + 1))

Index vs Value

∃𝑖 ∈ 1, … , 5 ∙ ∀𝑗 ∈ {1, … , 5} ∙ 𝑆(𝑖) ≥ 𝑆(𝑗) - There is an index i, and the value at that index is greater than or equal to every other value at every other index

Predicates Over Infinite Sequences

If a sequence is infinite, the last index and the values of terms beyond the ones given are unknown
We can make conjectures about subsequent values if we see a pattern
For S=[3,5,7,9…], ∀𝑖 ∈ ℕ ∖ {0} ∙ 𝑆(𝑖) = 2 ∗ 𝑖 + 1 can help predict the values of later terms

Fibonacci

The Fibonacci sequence is: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …
Each term is the sum of the two preceding terms
The predicate rule for the sequence: ∀𝑖 ∈ ℕ ∖ {0} ∙ 𝑆(𝑖) = 𝑆(𝑖 − 2) + 𝑆 𝑖 − 1

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Description

This quiz covers the concepts of predicates and quantifiers in mathematics, focusing on the existence and uniqueness of integers in various scenarios. It includes exercises on non-negative, even, odd, and prime integers, reinforcing the logical statements expressed through quantifiers.