Math 300 Questions

Let A={1,4,9} and B={1,3,6,10}. Find A\B.

{4,9}

Your friend tells you she has 7 coins in her hand (just pennies, nickels, dimes and quarters). If you guess how many of each kind of coin she has, she will give them to you. If you guess randomly, what is the probability that you will be correct?

(10C3)= 120

When playing Yahtzee, you roll five regular 6-sided dice. How many different outcomes are possible from a single roll? The order of the dice does not matter.

(10C5)

How many integer solutions to x1+x2+x3+x4=25 are there for which x1≥1, x2≥2, x3≥3, and x4≥4?

(18C15)

Conic, your favorite math themed fast food drive-in offers 20 flavors which can be added to your soda. You have enough money to buy a large soda with 4 added flavors. How many different soda concoctions can you order if: -You refuse to use any of the flavors more than once? -You refuse repeats but care about the order the flavors are added? -You allow yourself multiple shots of the same flavor? -You allow yourself multiple shots, and care about the order the flavors are added?

-(20C4) -P(20,4) -(23C4) -20^4

How many integer solutions are there to the equation x+y+z=8 for which -x, y, and z are all positive? -x, y, and z are all non-negative? -x, y, and z are all greater than or equal to −3.

-(7C2) -(10C2) -(19C2)

Decide whether it is an atomic statement, a molecular statement, or not a statement at all. The customers wore shoes and they wore socks.

Molecular, Conjunction

|A∩B| when A={x∈N:x≤20} and B={x∈N:x is prime}.

8

Find an example of sets A and B such that |A|=3, |B|=4, and |A∪B|=5.

A= {1,2,3} and B= {2,3,4,5}

Find an example of sets A and B such that |A|=4, |B|=5, and |A∪B|=9.

A={1,2,3,4} and B={5,6,7,8,9} gives A∪B={1,2,3,4,5,6,7,8,9}.

Let A={1,4,9} and B={1,3,6,10}. Find A∪B.

{1,3,4,6,9,10}

Decide whether it is an atomic statement, a molecular statement, or not a statement at all. The customers wore shoes.

Atomic

Describe a set using venn diagram with only A and B filled in.

A∪B\(A∩B)

Find the least element. {n∈N:n=k^2+1 for some k∈N}

1

Let A={1,2,...,10}. How many subsets of A contain exactly one element (i.e., how many singleton subsets are there)?

10

|A| when A={x∈Z:−2≤x≤100}

103

Let A={n∈N:20≤n<50} and B={n∈N:10<n≤30}.Suppose C is a set such that C⊆A and C⊆B. What is the largest possible cardinality of C?

11

If |A|=8 and |B|=5, what is |A∪B|+|A∩B|?

13

Let A={2,4,6,8}. Suppose B is a set with |B|=5. What are the smallest and largest possible values of |A×B|? Explain.

20, 20

For how many n∈{1,2,...,500} is n a multiple of one or more of 5, 6, or 7?

215

Your wardrobe consists of 5 shirts, 3 pairs of pants, and 17 bow ties. How many different outfits can you make?

255

Recall Z={...,−2,−1,0,1,2,...}(the integers). Let Z+={1,2,3,...} be the positive integers. Let 2Z be the even integers, 3Z be the multiples of 3, and so on. Find 2Z∩3Z. Describe the set in words, and using set notation.

2Z∩3Z is the set of all integers which are multiples of both 2 and 3 (so multiples of 6). Therefore 2Z∩3Z={x∈Z:∃y∈Z(x=6y)}.

Recall Z={...,−2,−1,0,1,2,...}(the integers). Let Z+={1,2,3,...} be the positive integers. Let 2Z be the even integers, 3Z be the multiples of 3, and so on. Express {x∈Z:∃y∈Z(x=2y∨x=3y)} as a union or intersection of two sets already described in this problem.

2Z∪3Z

Find the least element. {n∈N:n^2−3≥2}

3

Find the least element. {n∈N:n^2−5∈N}

3

|A| when A={4,5,6,...,37}.

33

How many 10-bit strings contain 6 or more 1's?

386

How many subsets of {0,1,...,9} have cardinality 6 or more?

386

A group of college students were asked about their TV watching habits. Of those surveyed, 28 students watch The Walking Dead, 19 watch The Blacklist, and 24 watch Game of Thrones. Additionally, 16 watch The Walking Dead and The Blacklist, 14 watch The Walking Dead and Game of Thrones, and 10 watch The Blacklist and Game of Thrones. There are 8 students who watch all three shows. How many students surveyed watched at least one of the shows?

39

Let A={1,2,...,10}. How many doubleton subsets (containing exactly two elements) are there?

45

Let A={1,2,3,4,5} and B={2,3,4}. How many sets C have the property that C⊆A and B⊆C.

4; {2,3,4}, {1,2,3,4}, {2,3,4,5}, and {1,2,3,4,5}

Let A={2,4,6,8}. Suppose B is a set with |B|=5. What are the smallest and largest possible values of |A∪B|? Explain.

5, 9

How many positive integers less than 1000 are multiples of 3, 5, or 7? Explain your answer using the Principle of Inclusion/Exclusion.

542

In a recent survey, 30 students reported whether they liked their potatoes Mashed, French-fried, or Twice-baked. 15 liked them mashed, 20 liked French fries, and 9 liked twice baked potatoes. Additionally, 12 students liked both mashed and fried potatoes, 5 liked French fries and twice baked potatoes, 6 liked mashed and baked, and 3 liked all three styles. How many students hate potatoes? Explain why your answer is correct.

6

Draw a Venn diagram. (A∪B)∖C

A, AB, and B filled in

Find an example of sets A and B such that A⊆B and A∈B.

A={1,2,3} and B={1,2,3,4,5,{1,2,3}}

Find an example of sets A and B such that A∩B={3,5} and A∪B={2,3,5,7,8}.

A={2,3,5,7,8} and B={3,5}

Let A2 be the set of all multiples of 2 except for 2. Let A3 be the set of all multiples of 3 except for 3. And so on, so that An is the set of all multiples of n except for n, for any n≥2. Describe (in words) the set A2∪A3∪A4∪⋯.

All Natural Numbers Except 0-3 and Prime Numbers

What is the coefficient of x^12 in (x+2)^15?

C(15,12)⋅2^3

Draw a Venn diagram. (A∩B)∪C

C, AC, BC, AB, and middle filled in

Draw a Venn diagram. A∩(B∪C)

Middle, AC, and AB filled in

Let A={x∈N:3≤x≤13}, B={x∈N:x is even}, and C={x∈N:x is odd}. Find B∪C.

Natural Numbers

Recall Z={...,−2,−1,0,1,2,...}(the integers). Let Z+={1,2,3,...} be the positive integers. Let 2Z be the even integers, 3Z be the multiples of 3, and so on. Is 2Z⊆Z+? Explain.

No

What is the coefficient of x^9 in the expansion of (x+1)^14+x^3(x+2)^15?

(14 + (15 (2^9) 9) 6)

A multiset is a collection of objects, just like a set, but can contain an object more than once (the order of the elements still doesn't matter). For example, {1,1,2,5,5,7} is a multiset of size 6. -How many sets of size 5 can be made using the 10 numeric digits 0 through 9? -How many multisets of size 5 can be made using the 10 numeric digits 0 through 9?

-(10C5) -(14C9)

A woman is getting married. She has 15 best friends but can only select 6 of them to be her bridesmaids, one of which needs to be her maid of honor. How many ways can she do this? -What if she first selects the 6 bridesmaids, and then selects one of them to be the maid of honor? -What if she first selects her maid of honor, and then 5 other bridesmaids? -Explain why 6(15C6)=15(14C5).

-(15C6)6 -15(14C5) -The first way gives the left-hand side of the identity and the second way gives the right-hand side of the identity. Therefore the identity holds.

After gym class you are tasked with putting the 14 identical dodgeballs away into 5 bins. -How many ways can you do this if there are no restrictions? -How many ways can you do this if each bin must contain at least one dodgeball?

-(18C4) -(13C4)

Using the digits 2 through 8, find the number of different 5-digit numbers such that: -Digits cannot be repeated and must be written in increasing order. -Digits can be repeated and must be written in non-decreasing order.

-(7C5) -(11C6)

A pizza parlor offers 10 toppings. -How many 3-topping pizzas could they put on their menu? Assume double toppings are not allowed. -How many total pizzas are possible, with between zero and ten toppings (but not double toppings) allowed? -The pizza parlor will list the 10 toppings in two equal-sized columns on their menu. How many ways can they arrange the toppings in the left column?

-120 -1024 -P(10,5)= 30240

You break your piggy-bank to discover lots of pennies and nickels. You start arranging these in rows of 6 coins. -You find yourself making rows containing an equal number of pennies and nickels. For fun, you decide to lay out every possible such row. How many coins will you need? -How many coins would you need to make all possible rows of 6 coins (not necessarily with equal number of pennies and nickels)?

-120 -384

Let S={1,2,3,4,5,6} -How many subsets are there of cardinality 4? -How many subsets of cardinality 4 have {2,3,5} as a subset? -How many subsets of cardinality 4 contain at least one odd number? -How many subsets of cardinality 4 contain exactly one even number?

-15 -3 -15 -3

Suppose you are ordering a large pizza from D.P. Dough. You want 3 distinct toppings, chosen from their list of 11 vegetarian toppings. -How many choices do you have for your pizza? -How many choices do you have for your pizza if you refuse to have pineapple as one of your toppings? -How many choices do you have for your pizza if you insist on having pineapple as one of your toppings? -How do the three questions above relate to each other? Explain.

-165 -120 -45 -165=120+45

Let S={1,2,3,4,5,6} -How many subsets are there total? -How many subsets have {2,3,5} as a subset? -How many subsets contain at least one odd number? -How many subsets contain exactly one even number?

-2^6=64 -2^3=8 -2^6−2^3=56 -3⋅2^3=24

Each of the counting problems below can be solved with stars and bars. For each, say what outcome the diagram ∗∗∗|∗||∗∗| represents, if there are the correct number of stars and bars for the problem. Otherwise, say why the diagram does not represent any outcome, and what a correct diagram would look like. -How many ways are there to select a handful of 6 jellybeans from a jar that contains 5 different flavors? -How many ways can you distribute 5 identical lollipops to 6 kids? -How many 6-letter words can you make using the 5 vowels in alphabetical order? -How many solutions are there to the equation x1+x2+x3+x4=6?

-3 strawberry, 1 lime, 0 licorice, 2 blueberry and 0 bubblegum. -This is backwards. ∗∗||∗∗∗||| first kid getting 2 lollipops, the third kid getting 3, and the rest of the kids getting none. -This is the word AAAEOO. -This doesn't represent a solution. An example of a correct diagram would be ∗|∗∗||∗∗∗ representing that x1=1, x2=2, x3=0, and x4=3.

We usually write numbers in decimal form (or base 10), meaning numbers are composed using 10 different "digits" {0,1,...,9}. Sometimes though it is useful to write numbers hexadecimal or base 16. Now there are 16 distinct digits that can be used to form numbers: {0,1,...,9,A,B,C,D,E,F}. -How many 2-digit hexadecimals are there in which the first digit is E or F? Explain your answer in terms of the additive principle (using either events or sets). -Explain why your answer to the previous part is correct in terms of the multiplicative principle (using either events or sets). Why do both the additive and multiplicative principles give you the same answer? -How many 3-digit hexadecimals start with a letter (A-F) and end with a numeral (0-9)? Explain. -How many 3-digit hexadecimals start with a letter (A-F) or end with a numeral (0-9) (or both)? Explain.

-32 -explanation -960 -3136

Consider all 5 letter "words" made from the letters a through h. -How many of these words are there total? -How many of these words contain no repeated letters? -How many of these words start with the sub-word "aha"? -How many of these words either start with "aha" or end with "bah" or both? -How many of the words containing no repeats also do not contain the sub-word "bad"?

-32768 words -6720 words -64 words -128 words -6660 words

How many lattice paths start at (3,3) and -end at (10,10)? -end at (10,10) and pass through (5,7)? -end at (10,10) and avoid (5,7)?

-3432 -840 - (14 − (6 (8 7) 2) 5)

Let A={1,2,3,...,9}. -How many subsets of A are there? That is, find |P(A)|. Explain. -How many subsets of A contain exactly 5 elements? Explain. -How many subsets of A contain only even numbers? Explain. -How many subsets of A contain an even number of elements? Explain.

-512 -126 -16 -256

How many 9-bit strings are there which: -Start with the sub-string 101? Explain. -Have weight 5 and start with the sub-string 101? Explain. -Either start with 101 or end with 11 (or both)? Explain. -Have weight 5 and either start with 101 or end with 11 (or both)? Explain.

-64 -20 -176 -51

Using the digits 2 through 8, find the number of different 5-digit numbers such that: -Digits can be used more than once. -Digits cannot be repeated, but can come in any order. -Digits cannot be repeated and must be written in increasing order. -Which of the above counting questions is a combination and which is a permutation? Explain why this makes sense.

-7^5= 16807 -P(7,5)= 2520 -21 -permutation= b, combination= c

For your college interview, you must wear a tie. You own 3 regular (boring) ties and 5 (cool) bow ties. -How many choices do you have for your neck-wear? -You realize that the interview is for clown college, so you should probably wear both a regular tie and a bow tie. How many choices do you have now? -For the rest of your outfit, you have 5 shirts, 4 skirts, 3 pants, and 7 dresses. You want to select either a shirt to wear with a skirt or pants, or just a dress. How many outfits do you have to choose from?

-8 ties -15 ties -42 outfits

Translate into English: -∀x(E(x)→E(x+2)) -∀x∃y(sin(x)=y) -∀y∃x(sin(x)=y) -∀x∀y(x^3=y^3→x=y).

-Any even number plus 2 is an even number. -For any x there is a y such that sin(x)=y. In other words, every number x is in the domain of sine. -For every y there is an x such that sin(x)=y. In other words, every number y is in the range of sine (which is false). -For any numbers, if the cubes of two numbers are equal, then the numbers are equal.

Let A={2,4,6,8}. Suppose B is a set with |B|=5. What are the smallest and largest possible values of |A∩B|? Explain.

0, 4

Find the least element. {n^2+1:n∈N}

1

Your Blu-ray collection consists of 9 comedies and 7 horror movies. Give an example of a question for which the answer is: -16. -63.

-For example, 16 is the number of choices you have if you want to watch one movie, either a comedy or horror flick. -For example, 63 is the number of choices you have if you will watch two movies, first a comedy and then a horror.

Consider the statement about a party, "If it's your birthday or there will be cake, then there will be cake." -Translate the above statement into symbols. Clearly state which statement is P and which is Q. -Make a truth table for the statement. -Assuming the statement is true, what (if anything) can you conclude if there will be cake? -Assuming the statement is true, what (if anything) can you conclude if there will not be cake? -Suppose you found out that the statement was a lie. What can you conclude?

-P: it's your birthday; Q: there will be cake. (P∨Q)→Q(P∨Q)→Q -three T's and one F. -Only that there will be cake. -It's NOT your birthday! -It's your birthday, but the cake is a lie.

Simplify the following statements (so that negation only appears right before variables). -¬(P→¬Q) -(¬P∨¬Q)→¬(¬Q∧R) -¬((P→¬Q)∨¬(R∧¬R)) -It is false that if Sam is not a man then Chris is a woman, and that Chris is not a woman.

-P∧Q -(¬P∨¬R)→(Q∨¬R) or, replacing the implication with a disjunction first: (P∧Q)∨(Q∨¬R) -(P∧Q)∧(R∧¬R) This is necessarily false, so it is also equivalent to P∧¬P -Either Sam is a woman and Chris is a man, or Chris is a woman.

Suppose P and Q are the statements: P: Jack passed math. Q: Jill passed math. -Translate "Jack and Jill both passed math" into symbols. -Translate "If Jack passed math, then Jill did not" into symbols. -Translate "P∨QP∨Q" into English. -Translate "¬(P∧Q)→Q¬(P∧Q)→Q" into English. *Suppose you know that if Jack passed math, then so did Jill. What can you conclude if you know that: Jill passed math? Jill did not pass math?

-P∧Q -P→¬Q -Jack passed math or Jill passed math (or both). -If Jack and Jill did not both pass math, then Jill did. *Nothing else. Jack did not pass math either.

We can also simplify statements in predicate logic using our rules for passing negations over quantifiers, and then applying propositional logical equivalence to the "inside" propositional part. Simplify the statements below (so negation appears only directly next to predicates). -There is a number n for which no other number is either less n than or equal to n. -It is false that for every number n there are two other numbers which n is between.

-There is a number n for which every other number is strictly greater than n. -There is a number n which is not between any other two numbers.

Let P(x) be the predicate, "4x+1 is even." -Is P(5) true or false? -What, if anything, can you conclude about ∃xP(x) from the truth value of P(5)? -What, if anything, can you conclude about ∀xP(x) from the truth value of P(5)?

-false -could be true or false -false

In my safe is a sheet of paper with two shapes drawn on it in colored crayon. One is a square, and the other is a triangle. Each shape is drawn in a single color. Suppose you believe me when I tell you that if the square is blue, then the triangle is green. What do you therefore know about the truth value of the following statements? -The square and the triangle are both blue. -The square and the triangle are both green. -If the triangle is not green, then the square is not blue. -If the triangle is green, then the square is blue. -The square is not blue or the triangle is green.

-false -not enough info -true -not enough info -true

Suppose the statement "if the square is blue, then the triangle is green" is true. Assume the converse is false. Classify each statement below as true or false (if possible). -The square is blue if and only if the triangle is green. -The square is blue if and only if the triangle is not green. -The square is blue. -The triangle is green.

-false -true -false -true

Consider the statement, "If you will give me a cow, then I will give you magic beans." Decide whether each statement below is the converse, the contrapositive, or neither. -If you will give me a cow, then I will not give you magic beans. -If I will not give you magic beans, then you will not give me a cow. -If I will give you magic beans, then you will give me a cow. -If you will not give me a cow, then I will not give you magic beans. -You will give me a cow and I will not give you magic beans. -If I will give you magic beans, then you will not give me a cow.

-neither -contrapositive -converse -neither -neither -neither

Classify each of the sentences below as an atomic statement, a molecular statement, or not a statement at all. If the statement is molecular, say what kind it is (conjunction, disjunction, conditional, biconditional, negation). -The sum of the first 100 odd positive integers. -Everybody needs somebody sometime. -The Broncos will win the Super Bowl or I'll eat my hat. -We can have donuts for dinner, but only if it rains. -Every natural number greater than 1 is either prime or composite. -This sentence is false.

-not a statement -atomic -molecular, disjunction -molecular, conditional -atomic -not a statement

Determine whether each molecular statement below is true or false, or whether it is impossible to determine. Assume you do not know what my favorite number is (but you do know that 13 is prime). -If 13 is prime, then 13 is my favorite number. -If 13 is my favorite number, then 13 is prime. -If 13 is not prime, then 13 is my favorite number. -13 is my favorite number or 13 is prime. -13 is my favorite number and 13 is prime. -7 is my favorite number and 13 is not prime. -13 is my favorite number or 13 is not my favorite number.

-not enough info -true -true -true -not enough info -false -true

Let P(x) be the predicate, "3x+1 is even." -Is P(5) true or false? -What, if anything, can you conclude about ∃xP(x) from the truth value of P(5)? -What, if anything, can you conclude about ∀xP(x) from the truth value of P(5)?

-true -true -could be true or false

Suppose you have sets A and B with |A|=10 and |B|=15. -What is the largest possible value for |A∩B|? -What is the smallest possible value for |A∩B|? -What are the possible values for |A∪B|?

-|A∩B|=10. -|A∩B|=0. -15≤|A∪B|≤25

Translate into symbols. Use E(x) for "x is even" and O(x) for "x is odd." -No number is both even and odd. -One more than any even number is an odd number. -There is prime number that is even. -Between any two numbers there is a third number. -There is no number between a number and one more than that number.

-¬∃x(E(x)∧O(x)) -∀x(E(x)→O(x+1)) -∃x(P(x)∧E(x)) (where P(x) means "x is prime"). -∀x∀y∃z(x<z<y∨y<z<x) -∀x¬∃y(x<y<x+1).

Recall Z={...,−2,−1,0,1,2,...}(the integers). Let Z+={1,2,3,...} be the positive integers. Let 2Z be the even integers, 3Z be the multiples of 3, and so on. Is Z+⊆2Z? Explain.

No

Are there sets A and B such that |A|=|B|, |A∪B|=10, and |A∩B|=5? Explain.

No, cardinality of A is the same as cardinality of B and if the union cardinality is 10, then they will have intersection at all 10 terms.

Decide whether it is an atomic statement, a molecular statement, or not a statement at all. Customers must wear shoes.

Not a statement

Draw a Venn diagram. A[complement]∩B∩C[complement]

Only B filled in

Draw a Venn diagram. (A∪B)[complement]

Only background filled in

A combination lock consists of a dial with 40 numbers on it. To open the lock, you turn the dial to the right until you reach a first number, then to the left until you get to second number, then to the right again to the third number. The numbers must be distinct. How many different combinations are possible?

P(40,3)= 40x39x38= 59280

Let A={a,b,c,d}. Find P(A).

P(A)={∅,{a},{b},{c},{d},{a,b},{a,c},{a,d},{b,c},{b,d},{c,d}{a,b,c},{a,b,d},{a,c,d},{b,c,d},{a,b,c,d}}.

Give a combinatorial proof of the identity 2+2+2=3⋅2.

Question: How many 2-letter words start with a, b, or c and end with either y or z? Answer 1: There are two words that start with a, two that start with b, two that start with c, for a total of 2+2+2. Answer 2: There are three choices for the first letter and two choices for the second letter, for a total of 3⋅2. Since the two answers are both answers to the same question, they are equal. Thus 2+2+2=3⋅2.

Give a combinatorial proof of the identity (nC2)(n−2Ck−2)=(nCk)(kC2).

Question: You have a large container filled with ping-pong balls, all with a different number on them. You must select kk of the balls, putting two of them in a jar and the others in a box. How many ways can you do this? Answer 1: First select 2 of the n balls to put in the jar. Then select k−2 of the remaining n−2 balls to put in the box. The first task can be completed in (nC2) different ways, the second task in (n−2Ck−2) ways. Thus there are (nC2)(n−2Ck−2) ways to select the balls. Answer 2: First select k balls from the n in the container. Then pick 2 of the k balls you picked to put in the jar, placing the remaining k−2 in the box. The first task can be completed in (nCk) ways, the second task in (kC2) ways. Thus there are (nCk)(kC2) ways to select the balls. Since both answers count the same thing, they must be equal and the identity is established.

Determine if the following deduction rule is valid: P∨Q ¬P ∴Q

The deduction rule is valid. To see this, make a truth table which contains P∨Q and ¬P (and P and Q of course). Look at the truth value of Q in each of the rows that have P∨Q and ¬P true.

Draw a Venn diagram. A∪B[complement]

Whole thing filled in except only B

Make a truth table for the statement ¬P∧(Q→P). What can you conclude about P and Q if you know the statement is true?

false, false, false, true

Make a truth table for the statement (P∨Q)→(P∧Q).

true, false, false, true

Let A={x∈N:3≤x≤13}, B={x∈N:x is even}, and C={x∈N:x is odd}. Find B∩C.

{ }

Let A={1,2,3,4,5}, B={3,4,5,6,7}, and C={2,3,5}. Find A∪B.

{1,2,3,4,5,6,7}

Find a set of smallest possible size that has both {1,2,3,4,5} and {2,4,6,8,10} as subsets.

{1,2,3,4,5,6,8,10}

Let A={1,2,3,4,5}, B={3,4,5,6,7}, and C={2,3,5}. Find A∖B.

{1,2}

Let A={1,2,3,4,5}, B={3,4,5,6,7}, and C={2,3,5}. Find A∩[(B∪C)]complement.

{1}

Let A={1,4,9} and B={1,3,6,10}. Find A∩B.

{1}

Find a set of largest possible size that is a subset of both {1,2,3,4,5} and {2,4,6,8,10}.

{2,4}

Let A={x∈N:3≤x≤13}, B={x∈N:x is even}, and C={x∈N:x is odd}. Find A∪B.

{3,4,5,6,7,8,9,10,11,12,13}

Let A={1,2,3,4,5}, B={3,4,5,6,7}, and C={2,3,5}. Find A∩B.

{3,4,5}

Let A={1,4,9} and B={1,3,6,10}. Find B\A.

{3,6,10}

Let A={x∈N:3≤x≤13}, B={x∈N:x is even}, and C={x∈N:x is odd}. Find A∩B.

{4,6,8,10,12}

Let A={x∈N:4≤x<12} and B={x∈N:x is even}. Find A∩B.

{4,6,8,10}

Let A={x∈N:4≤x<12} and B={x∈N:x is even}. Find A∖B.

{5,7,9,11}

Let A={1,2,3,4,5,6}. Find all sets B∈P(A) which have the property {2,3,5}⊆B.

{{2,3,5}, {1,2,3,5}, {1,2,3,4,5}, {1,2,3,4,5,6}, {2,3,4,5,6}, {2,3,4,5}, {2,3,5,6}, {1,2,3,5,6}}