question archive Question #1) For each of (a), (b), (c): (a): ∀m ∈ Z ?n ∈ Z m + n = 10 (b): ?m ∈ N ∀n ∈ Z (n ≤ m ∨ n > 2m) (c): ?m ∈ Z ∀n ∈ Z ?p ∈ Z 2p − n ≤ m Replace each ? with either the universal quantifier (∀) or the existential qualifier (∃) such that the resulting statement is true
Question #1) For each of (a), (b), (c): (a): ∀m ∈ Z ?n ∈ Z m + n = 10 (b): ?m ∈ N ∀n ∈ Z (n ≤ m ∨ n > 2m) (c): ?m ∈ Z ∀n ∈ Z ?p ∈ Z 2p − n ≤ m Replace each ? with either the universal quantifier (∀) or the existential qualifier (∃) such that the resulting statement is true
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Question #1)
For each of (a), (b), (c):
- (a): ∀m ∈ Z ?n ∈ Z m + n = 10
- (b): ?m ∈ N ∀n ∈ Z (n ≤ m ∨ n > 2m)
- (c): ?m ∈ Z ∀n ∈ Z ?p ∈ Z 2p − n ≤ m
- Replace each ? with either the universal quantifier (∀) or the existential qualifier (∃) such that the resulting statement is true. Note that the replacement of ? with quantifiers is the only modification allowed. (The order of the quantifiers cannot be changed. "Z" = integers and "N" = natural numbers)
- Write out the resulting statement in common English.
- Explain informally but rigorously why the statement is true.
Question #2)
There is a coin collection.
- The coin display case contains 441 slots arranged in a 21 X 21 grid. Each slot can only hold one coin. There are 21 pennies. Each penny is indistinguishable from the others. These 21 pennies must be arranged such that no two pennies are in the same row or column. How many ways can the coins be arranged?
- There is another 21 X 21 display case with 441 slots same as above. There are 21 quarters. The quarters are all distinct, each has a different state on the back. These 21 quarters must be arranged such that no two quarters are in the same row or column. How many ways can the coins be arranged?
