Practice Midterm 1
... (b) Suppose that 3 was a positive rational number. Let us say that 3 is equal to c/d, where c and d are natural numbers. Use part 4(a) to show that 3d − c < c. ...
... (b) Suppose that 3 was a positive rational number. Let us say that 3 is equal to c/d, where c and d are natural numbers. Use part 4(a) to show that 3d − c < c. ...
Lecture 6: End and cofinal extensions
... are equivalent for all n ∈ N. (a) M 4n+1 K. (b) For every η(x̄) ∈ Πn (M ), if K |= ∃x̄ η(x̄), then K |= η(c̄) for some c̄ ∈ M . Proof. The implication (a) ⇒ (b) is straightforward. We prove the converse implication (b) ⇒ (a) by strong induction on n. Let n ∈ N for which (b) is true, and (b) ⇒ (a) fo ...
... are equivalent for all n ∈ N. (a) M 4n+1 K. (b) For every η(x̄) ∈ Πn (M ), if K |= ∃x̄ η(x̄), then K |= η(c̄) for some c̄ ∈ M . Proof. The implication (a) ⇒ (b) is straightforward. We prove the converse implication (b) ⇒ (a) by strong induction on n. Let n ∈ N for which (b) is true, and (b) ⇒ (a) fo ...
Lecture notes for Section 8.2
... 4. If b2 – 4ac is negative, then there are two solutions that are complex and unequal. Example: 4 x 2 2 x 9 0 ...
... 4. If b2 – 4ac is negative, then there are two solutions that are complex and unequal. Example: 4 x 2 2 x 9 0 ...
