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Questions and Answers
What characterizes an ordered list compared to a sorted list?
What characterizes an ordered list compared to a sorted list?
For the sequence S = [4, 13, 72, 89, 51], what does the predicate ∃𝑖 ∈ {1, … , 5} ∙ 𝑆(𝑖) = 89 indicate?
For the sequence S = [4, 13, 72, 89, 51], what does the predicate ∃𝑖 ∈ {1, … , 5} ∙ 𝑆(𝑖) = 89 indicate?
Which sequence is described as sorted in ascending order?
Which sequence is described as sorted in ascending order?
Which statement accurately describes finite sequences?
Which statement accurately describes finite sequences?
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What does the notation ∀𝑖 ∈ {1, … , 4} ∙ 𝑆(𝑖) < 𝑆(𝑖 + 1) represent?
What does the notation ∀𝑖 ∈ {1, … , 4} ∙ 𝑆(𝑖) < 𝑆(𝑖 + 1) represent?
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In the context of sequences, what does the term 'index' refer to?
In the context of sequences, what does the term 'index' refer to?
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Which of the following is true about an infinite sequence?
Which of the following is true about an infinite sequence?
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What is the primary difference between the sequences [15, 41, 89] and [41, 89, 15]?
What is the primary difference between the sequences [15, 41, 89] and [41, 89, 15]?
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What does the statement ∃𝑥∈𝐷·𝑤𝑜𝑚𝑎𝑛 𝑥 ⇒ 𝑡𝑎𝑙𝑙(𝑥) imply?
What does the statement ∃𝑥∈𝐷·𝑤𝑜𝑚𝑎𝑛 𝑥 ⇒ 𝑡𝑎𝑙𝑙(𝑥) imply?
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How would you interpret the predicate ∃𝑥∈𝐴·∃𝑦∈𝐵·𝑥 = 𝑦?
How would you interpret the predicate ∃𝑥∈𝐴·∃𝑦∈𝐵·𝑥 = 𝑦?
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What does the statement ∀𝑥∈𝐴·∃𝑦∈𝐵·𝑥 < 𝑦 mean?
What does the statement ∀𝑥∈𝐴·∃𝑦∈𝐵·𝑥 < 𝑦 mean?
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What is a consequence of the statement ∃𝑥∈𝐷·𝑤𝑜𝑚𝑎𝑛 𝑥 ∧ 𝑡𝑎𝑙𝑙(𝑥)?
What is a consequence of the statement ∃𝑥∈𝐷·𝑤𝑜𝑚𝑎𝑛 𝑥 ∧ 𝑡𝑎𝑙𝑙(𝑥)?
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What is the implication of using the symbol 'L(a, b)', where L stands for love?
What is the implication of using the symbol 'L(a, b)', where L stands for love?
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What logical notation represents 'There exists an even integer'?
What logical notation represents 'There exists an even integer'?
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How would you express 'Every integer is even or odd' in logical notation?
How would you express 'Every integer is even or odd' in logical notation?
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What does the notation ∀𝑥∈ℤ: 𝐸(𝑥) ∧ 𝑃(𝑥) ⇒ 𝑥 = 2 represent?
What does the notation ∀𝑥∈ℤ: 𝐸(𝑥) ∧ 𝑃(𝑥) ⇒ 𝑥 = 2 represent?
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Which of the following represents 'Not all integers are odd'?
Which of the following represents 'Not all integers are odd'?
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How can 'Not all primes are odd' be expressed logically?
How can 'Not all primes are odd' be expressed logically?
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Which logical statement correctly states that all prime integers are non-negative?
Which logical statement correctly states that all prime integers are non-negative?
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What does the statement 'If an integer is not odd, then it must be even' represent in logical notation?
What does the statement 'If an integer is not odd, then it must be even' represent in logical notation?
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Which notation shows that not every person in the class csc1026 gets the train?
Which notation shows that not every person in the class csc1026 gets the train?
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What is the correct representation of 'Everyone in csc1026 has lectures on Thursday and Friday'?
What is the correct representation of 'Everyone in csc1026 has lectures on Thursday and Friday'?
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What does the logical statement ∃𝑥∈ℤ: 𝑃(𝑥) ∧ ¬𝑂(𝑥) convey?
What does the logical statement ∃𝑥∈ℤ: 𝑃(𝑥) ∧ ¬𝑂(𝑥) convey?
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What condition does the predicate ∀𝑖 ∈ {1, … , 4} ∙ (𝑆(𝑖) ≤ 𝑆(𝑖 + 1)) ∨ (𝑆(𝑖) ≥ 𝑆(𝑖 + 1)) express for a sequence?
What condition does the predicate ∀𝑖 ∈ {1, … , 4} ∙ (𝑆(𝑖) ≤ 𝑆(𝑖 + 1)) ∨ (𝑆(𝑖) ≥ 𝑆(𝑖 + 1)) express for a sequence?
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Which example satisfies the predicate ∀𝑖 ∈ {1, … , 4} ∙ (𝑆(𝑖) ≤ 𝑆(𝑖 + 1)) ∨ (𝑆(𝑖) ≥ 𝑆(𝑖 + 1)) but is not fully sorted?
Which example satisfies the predicate ∀𝑖 ∈ {1, … , 4} ∙ (𝑆(𝑖) ≤ 𝑆(𝑖 + 1)) ∨ (𝑆(𝑖) ≥ 𝑆(𝑖 + 1)) but is not fully sorted?
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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?
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?
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What does the predicate ∀𝑖 ∈ ℕ ∖ {0} ∙ 𝑆(𝑖) = 2 ∗ 𝑖 + 1 represent?
What does the predicate ∀𝑖 ∈ ℕ ∖ {0} ∙ 𝑆(𝑖) = 2 ∗ 𝑖 + 1 represent?
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Which correctly describes the Fibonacci sequence?
Which correctly describes the Fibonacci sequence?
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Why does the predicate ∀𝑖 ∈ {1, … , 4} ∙ (𝑆(𝑖) ≤ 𝑆(𝑖 + 1)) ∨ (∀𝑗 ∈ {1, … , 4} ∙ 𝑆(𝑗) ≥ 𝑆(𝑗 + 1)) need to be modified?
Why does the predicate ∀𝑖 ∈ {1, … , 4} ∙ (𝑆(𝑖) ≤ 𝑆(𝑖 + 1)) ∨ (∀𝑗 ∈ {1, … , 4} ∙ 𝑆(𝑗) ≥ 𝑆(𝑗 + 1)) need to be modified?
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To determine potential subsequent values in an observed pattern of an infinite sequence, what is essential?
To determine potential subsequent values in an observed pattern of an infinite sequence, what is essential?
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Which of the following is true about infinite sequences discussed?
Which of the following is true about infinite sequences discussed?
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What does the expression ∀𝑥∈𝐷∃𝑦∈𝐷𝐿(𝑦,𝑥) signify?
What does the expression ∀𝑥∈𝐷∃𝑦∈𝐷𝐿(𝑦,𝑥) signify?
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What is the negation of the statement 'All dogs have fleas' expressed in logical terms?
What is the negation of the statement 'All dogs have fleas' expressed in logical terms?
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Which expression means 'Someone loves someone' in logical notation?
Which expression means 'Someone loves someone' in logical notation?
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What is the logical implication of the statement '∀𝑥∈𝐷𝐿(𝑥,𝑥)'?
What is the logical implication of the statement '∀𝑥∈𝐷𝐿(𝑥,𝑥)'?
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Which statement represents 'Someone is loved by everyone' logically?
Which statement represents 'Someone is loved by everyone' logically?
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What is the correct negation of the statement 'Some drivers do not obey the speed limit'?
What is the correct negation of the statement 'Some drivers do not obey the speed limit'?
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Which logical expression represents 'Everyone loves everyone'?
Which logical expression represents 'Everyone loves everyone'?
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What is the outcome of applying De Morgan's rule to ¬∃𝑥∈𝐷∀𝑦∈𝐷∀𝑧∈𝐷𝑃(𝑥,𝑦,𝑧)?
What is the outcome of applying De Morgan's rule to ¬∃𝑥∈𝐷∀𝑦∈𝐷∀𝑧∈𝐷𝑃(𝑥,𝑦,𝑧)?
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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 ∃𝑥𝑥 2 = (𝑥 ∗ 2)
- The integer that satisfies the statement is 0
Quantifiers
- ∀𝑥𝐷𝑃 𝑥 is the same as: 𝑃 1 ∧ 𝑃(2) ∧ 𝑃(3) when D = { 1, 2, 3 }
- ∃𝑥𝐷𝑃 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
- ∀𝑥𝐷𝑥 𝜖 𝑐𝑠𝑐1026 ⇒ 𝑃 𝑥 - Everyone in csc1026 has lectures on Thurs & Fri
- ∃𝑥𝐷𝑥 𝜖 𝑐𝑠𝑐1026 ∧ 𝑄(𝑥) - Some people in csc1026 get the train to come to university
- ∃𝑥𝐷 𝑤𝑜𝑚𝑎𝑛 𝑥 ∧ 𝑡𝑎𝑙𝑙(𝑥) - There exists a tall woman
Combining Quantifiers
- Let A = {2, 6, 14} and B = {6, 8, 11}
- ∃𝑥𝐴 ∃𝑦𝐵 𝑥 = 𝑦 - This means: "There exists an x in A such that there exists a y in B such that x = y."
- ∀𝑥𝐴 ∃𝑦𝐵 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”
- ∀𝑥𝐷 ∃𝑦𝐷 𝐿 𝑦, 𝑥 - Everyone is loved by someone
- ∀𝑥𝐷 ∃𝑦𝐷 𝐿 𝑥, 𝑦 - Everyone loves someone
- ∃𝑥𝐷 ∀𝑦𝐷 𝐿 𝑥, 𝑦 - Someone loves everyone
- ∃𝑥𝐷 ∀𝑦𝐷 𝐿 𝑦, 𝑥 - Someone is loved by everyone
- ∃𝑥𝐷 𝐿 𝑥, 𝑥 - Someone loves them self
- ∀𝑥𝐷 𝐿 𝑥, 𝑥 - Everyone loves themselves
- ∃𝑥𝐷 ∃𝑦𝐷 𝐿 𝑥, 𝑦 - Someone loves someone
- ∃𝑥𝐷 ∃𝑦𝐷 𝐿 𝑦, 𝑥 - Someone is loved by someone
- ∀𝑥𝐷 ∀𝑦𝐷 𝐿 𝑥, 𝑦 - Everyone loves everyone
- ∀𝑥𝐷 ∀𝑦𝐷 𝐿 𝑦, 𝑥 - Everyone is loved by everyone
Exercise
- "Some drivers do not obey the speed limit" - ∃𝑥𝑑𝑟𝑖𝑣𝑒𝑟𝑠 ¬𝑜𝑏𝑒𝑦𝐿𝑖𝑚𝑖𝑡(𝑥)
- "All dogs have fleas" - ∀𝑥𝑑𝑜𝑔𝑠 𝑓𝑙𝑒𝑎𝑠 𝑥
Negation of Multiple Quantifiers
- Basic rule for negation of quantified predicates: Flip the quantifier and negate the predicate
- Example: ¬∃𝑥𝐷∀𝑦𝐷∀𝑧𝐷𝑃(𝑥, 𝑦, 𝑧) = ∀𝑥𝐷 ∃𝑦𝐷 ∃𝑧𝐷 ¬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.