Symbols & Definitions
What does ㄱ symbol mean?
false/not
An or statement is the same as:
Unions.
What are the two aspects to proving statements? Explain.
Analysis (bottom-up)
hypothesis
conclusion
justification
Synthesis (top-down)
When can you use theorems?
When the hypotheses of the theorem match the hypotheses and variables of the statement you are trying to prove.
The abbreviation for proving for all statements.
(pr. ∀)
What does ⊆ symbol mean?
subset
The two steps before proving this statement are:
t ∈ C and x ∈ C
t ∈ C
x ∈ C
Define X = {t ∈ N | 9 < t}
b ∈ X
_______. (def. X)
9 < b
What is the difference between theorems and axiom?
Theorems are proven true while axioms are assumed true.
Name 4 types of justifications. ex. (2; ____)
Any four below:
transitivity, hypothesis, definition, subset, given, arbitrary, and, or, for all
What is a counterexample?
An example that proves that something is false.
Using the method of proving or statements, write the steps for:
x ≤ 9 or x ∈ C
Case 1:
x ∉ C
.
.
.
x ≤ 9
Case 2:
x > 9
.
.
.
x ∈ C
Let S and T be arbitrary sets. Start to develop a proof that S ⊆ T and write as many steps as you can using this information. (Do not make up new information; you don't know anything about sets S and T).
Provide a counterexample to show that the conclusion is not true.
Let x ∈ S be arbitrary
x ∈ T
For all x ∈ S : x ∈ T (pr. ∀)
S ⊆ T (def. ⊆)
Counterexample:
If S={1, 3, 5, 7}
T={2, 4, 6, 8}
S ⊆ T is not true
The hypotheses and the conclusion of the Mean Value Theorem are:
Hypotheses:
f(x) is a continuous function on the closed interval [a, b].
f(x) is a differentiable function on the open interval (a, b)
Conclusion:
There is at least one number c where a < c < b such that f'(c)= (f(b)-f(c))/(b-a)
What is the justifications for this example?
1. x < 5
2. y < 7
3. 5 < 7
4. x < 7 (______;_______)
(1, 3; Trans. <)
What are the symbols and definitions of union and intersections?
Union: ∪ , A or B
Intersection: ∩ , A and B
Draw the Venn diagram for or, and, for allstatements.
(drawn on board)
Prove the following:
Let A = {a ∈ N | a < 10} and B = {b ∈ N | 5 < b}. Then A ⊆ B.
Let x ∈ A be arbitrary
x < 10 (def. A)
5 < 10 (def. trans.)
5 < x < 10 (def. trans.)
x ∈ B (def. B)
For all x ∈ A : x ∈ B
A ⊆ B
Prove Theorem 9.1:
For sets A and B
a) A ∩ B ⊆ A
b) A ∩ B ⊆ B
b) Let x ∈ A ∩ B
x ∈ A and x ∈ B. (def. ∩)
x ∈ B (us. &)
for all x ∈ A ∩ B : x ∈ B (pr. ∀)
A ∩ B ⊆ B (def. ⊆)
a)Let x ∈ A ∩ B
x ∈ A and x ∈ B. (def. ∩)
x ∈ A (us. &)
for all x ∈ A ∩ B : x ∈ A (pr. ∀)
A ∩ B ⊆ A (def. ⊆)
What is the justification for this example?
1. S ⊆ T
2. for all y ∈ S : y ∈ T (______;_______)
(1, def. ⊆)
What does this symbol stand for ∀ and what is it's definition?
Arbitrary: when a chosen element about which we assume nothing except that it is in the set is called an arbitrary element.
Assume
A ∪ B ⊆ A ∪ C
Prove
B ⊆ C
A ∪ B ⊆ A ∪ C
for all A ∪ B : A ∪ C. (def. ⊆)
Let x ∈ A be arbitrary
x ∈ A or x ∈ B (def.∪)
Assume x ∉ A
x ∈ B (us. or)
x ∈ A or x ∈ C. (def.∪)
Assume x ∉ A
x ∈ C (us. or)
for all x ∈ B : x ∈ C (pr. ∀)
B ⊆ C (def. ⊆)
Prove the following statement:
Assume K = { x ∈ N | x < 20 }
J ⊆ K
Show for all a ∈ J : a < 20
Let a ∈ J be arbitrary
J ⊆ K. (hyp)
for all a ∈ J : a ∈ K. (pr. ∀)
a ∈ K (us. ∀)
a < 20 (def. K)
for all a ∈ J : a < 20 (pr. ∀)
Let A, B, and C be sets. Prove that if A is a subset of (B intersection C), then A is a subset of B. Use Theorems 5.1 and 9.1 not definitions.
9.1:
For sets A and B:
A ∩ B ⊆ A
A ∩ B ⊆ B
5.1:
For sets A, B, and C, if A ⊆ B and B ⊆ C, then A ⊆ C.
A, B, and C are sets
A ⊆ B ∩ C
B ∩ C ⊆ B (theorem 9.1)
A ⊆ B ∩ C ⊆ B (us. ∀, imp.)
A ⊆ B (theorem 5.1)
What is the justification for this example?
1. Let t ∈ S be arbitrary.
.
7. 13 < t
8. for all t ∈ S : 13 < t (_______;_________)
(1—7; pr. ∀)