Proof and Mathematical Communication

Question 1

1.)What is an identity?
2.)What symbol is used to display an identity?
3.)What do you call two statements connected by an identity symbol?

Answer 1

1.)An identity is a relation that is true for for all values of the variable
2.) ≡ or =
3.) Congruent expressions

Question 2

1.) what does the ⇒ symbol mean?
2.) what does the ⇐ symbol mean?
3.) what does the ⇔ symbol mean?

Answer 2

1.) the subsequent statement follows from the previous statement but is not mathematically identical
2.) the previous statement follows from the subsequent statement
3.) a subsequent statement is equivalent to the previous one

Question 3

What 2 operations can change the number of solutions an equation has ?

Answer 3

÷ 0 and squaring equations.

Question 4

convert the following inequalities into interval notation:
1.) a < x < b
2.) a ⩽ x ⩽ b
3.) a ⩽ x < b
4.) a < x ⩽ b
5.) a < x
6.) a ⩽ x
7.) a > x
8.) a ⩾ x

Answer 4

1.) x ∈ (a, b)
2.) x ∈ [a, b]
3.) x ∈ [a, b)
4.) x ∈ (a, b]
5.) x ∈ (-∞ ,a)
6.) x ∈ (- ∞ ,a]
7.) x ∈ (a,∞ )
8.) x ∈ [a,∞ )

Question 5

convert the following inequalities into set notation:
1.) a < x < b
2.) a ⩽ x ⩽ b
3.) a ⩽ x < b
4.) a ⩽ x ⩽ b
5.) a < x
6.) a ⩽ x
7.) a > x
8.) a ⩾ x

Answer 5

1.) {x : a < x < b}
2.) {x : a ⩽ x ⩽ b}
3.) {x : a ⩽ x < b}
4.) {x : a ⩽ x ⩽ b}
5.) {x : a < x}
6.) {x : a ⩽ x}
7.) {x : a > x}
8.) {x : a ⩾ x}

Question 6

If there is no solution to the inequality, what do we write?

Answer 6

x ∈ ∅

Question 7

1.) what does A ⋃ B describe?
2.) what does A ⋂ B describe?

Answer 7

1.) the union of A and B, It means that the solution is in A or B or both
2.) The intersection of A and B, It means that the solution lies in both A and B

Question 8

When trying to disprove by counter example, what are good number to test in general?

Answer 8

0, negative integers or large values

Question 9

1.)How do you describe an even number in algebraic form?
2.) How do you describe an odd number in algebraic form?

Answer 9

1.) 2n where n is an integer
2.) 2n+1where n is an integer

Question 10

When proving by deduction to see if a number is some multiple of the number n, what form should the final result be put in?

Answer 10

n(f(k))

Question 11

When proving if a number p is prime, what integer n do we need to check for divisibility for

Answer 11

n=√p

Question 12

When proving by exhaustion what needs to be done to prove such a statement

Answer 12

all values in the set used must be tested and match the conjecture