Sets and Proofs

Set Theory 1

100

What does the set Z represent?

Integers

200

Of the following statements, which is true?

Q is a subset of R

R is a subset of Q

Q is a subset of R

300

Give me the smallest number of the interval [-4, 5)

-4

400

Given sets A and B, how can we show that A is a subset of B?

If for every element x in A, x is in B.

500

Give me the largest number in the interval [-4, 5)

Trick question, there is no answer

Set Theory 2

100

Let set A = {1, 2, 3}

Let set B = {a, b, c}

What is A U B?

A U B = {1, 2, 3, a, b, c}

200

Let set A = {1, 2, 3}

Let set B = {2, 3, 4}

What is A intersect B?

A intersect B = {2, 3}

300

For any set A, is A a proper subset of A?

No

400

What is the symbol of the following set?

Z⁺ U {0}

N

500

Let set A be {a, b, c}

What is |P(A)|?

8

Set Theory 3

100

Give a example of a set which the empty set is a subset of. 

Literally any set

200

Let set A = {1, 2, 3}

Let set B = {a, b}

What are the elements of A x B?

A x B = {(1, a), (1, b), (2, a), (2, b), (3, a), (3, b)}

300

Prove, using a Venn Diagram, the equivalence of the following:

A \ B = A intersect (compliment of B)

Solution drawn on board

400

Suppose 120 people signed up for a running competition, and 85 of them identified themselves as a short-distance runner, and 65 of them identified themselves as a long-distance runner, how many identified as both?

85+65 - 120 = 30 identified as both a long-distance and a short-distance runner

500

What could be U (universal set) in the context of Z is a subset of U if Z denotes the set of all integers.

Z, Q, R, or any set that contains the set of integers. 

Proofs

100

Prove that an odd integer plus an even integer is odd. 

Let a = 2x

Let b = 2y + 1

x, y are integers

then a + b = 2x + 2y + 1 = 2(x+y) + 1

which is by definition odd. 

200

Prove that the sum of 2 consecutive integers are odd.

Solution written on board. 

300

Suppose a² is odd, show that a is odd. 

Prove by contraposition. Solution written on board. 

400

Suppose that you know that b is odd, and ab is also odd. Show by contradiction that a must be odd. 

Suppose b = 2n+1, and ab = 2x + 1

Let a = 2m, then ab = 2m(2n+1) = 4mn + 2m = 2(2mn + m)

Thus ab is by definition even, but the premise is that it is odd, therefore we arrive at a contradiction.

500

Show that the square root of 2 is irrational.

Proof by contradiction. Solution written on board.