Basics of math 1 Logic and sets The statement a is true, b is false. Both statements are false. 9000086601 (level 1): Let a and b be two sentences in the sense of mathematical logic. It is known that the composite statement 9000086604 (level 1): Let a and b be two sentences in the sense of mathematical :(a _ b) logic. It is known that the composite statement is true. For each a and b determine whether it is true or false. :(a ^ :b) is false. For each a and b determine whether it is true or false. Both statements are false. The statement a is true, b is false. Both statements are true. The statement a is true, b is false. Both statements are true. The statement a is false, b is true. The statement a is false, b is true. Both statements are false. 9000086602 (level 1): Let a and b be two sentences in the sense of mathematical logic. It is known that the composite statement 9000086605 (level 1): Let a and b be two sentences in the sense of mathematical :a _ b logic. It is known that the composite statement is false. For each a and b determine whether it is true or false. :a =):b The statement a is true, b is false. is false. For each a and b determine whether it is true or false. The statement a is false, b is true. Both statements are true. The statement a is false, b is true. Both statements are true. The statement a is true, b is false. Both statements are false. Both statements are false. 9000086603 (level 1): Let a and b be two sentences in the sense of mathematical logic. It is known that the composite statement 9000086606 (level 1): Let a and b be two sentences in the sense of mathematical :a ^ b logic. It is known that the composite statement is true. For each a and b determine whether it is true or false. a () (a _ b) The statement a is false, b is true. is false. For each a and b determine whether it is true or false. The statement a is false, b is true. Both statements are true. Both statements are true. 1 The statement a is true, b is false. a () (b _ c) (:a _ b) _ c (a ^ b) _ c Both statements are false. (a _ b) =):c 9000086607 (level 1): 9000080901 (level 2): Let a and b be two sentences in the sense of mathematical Find the intersection A \ B for A = {−5; 0; 1:5; 2; 6g and logic. It is known that the composite statement B = fx 2 Z; x ≥ 0g. (:a _ b) ^ a f0; 2; 6g f0; 1:5; 2; 6g f1:5; 2; 6g Z is true. For each a and b determine whether it is true or false. Both statements are true. 9000080902 (level 2): Find the intersection A \ B for A = fx 2 Z; x ≥ −2g and The statement a is true, b is false. B = fx 2 N; x ≤ 5g. The statement a is false, b is true. f1; 2; 3; 4; 5g f0; 1; 2; 3; 4; 5g Both statements are false. f0; 1; 2; 3; 4g {−2; −1; 0; 1; 2; 3; 4; 5g 9000086608 (level 1): 9000080903 (level 2): Let a and b be two sentences in the sense of mathematical Find the union A [ B for A = fx 2 Z; x ≥ −3g and logic. It is known that the composite statement B = fx 2 N; x < 8g. :a () (a ^ b) fx 2 Z; x ≥ −3g is true. For each a and b determine whether it is true or false. f1; 2; 3; 4; 5; 6; 7g The statement a is true, b is false. {−3; −2; −1; 0; 1; 2; 3; 4; 5; 6; 7g Both statements are true. Z The statement a is false, b is true. Both statements are false. 9000080904 (level 2): Find the union A [ B for A = N and B = fx 2 Z; x > 8g. 9000086609 (level 1): ; fx 2 ; x > 8g Let the sentence a be true and the sentences b and c be false. N Z In the following list identify a true statement. Z (a _ b) =):c (:a _ b) _ c (a ^ b) _ c 9000080905 (level 2): a () (b _ c) Identify the set B which satisfies A [ B = C 9000086610 (level 1): Let the sentences a and b be false and c be true. In the if A = fx 2 N; x < 3g and C = f0; 1; 2g. following list identify a false statement. 2 f0; 1; 2g; f0; 1g; f0; 2g; f0g 9000089001 (level 3): Students from a class have a possibility to work in mathematical and physical hobby groups. no solution exist • There are 31 students in the class. • From the total, 21 students are members of mathematical ; group. • Some of the students are members of both groups, but f0; 1; 2g; f0; 1g; f1; 2g; f0; 2g there are 10 which are members of just one group. • There are 3 students which are not members of any of those groups. 9000080906 (level 2): How many students are members of both mathematical and 0 Find BA (the complement to B in A) for A = fx 2 N; x < 9g physical groups? and B = f4; 5; 6; 7g. 18 16 19 f1; 2; 3; 8g ; f4; 5; 6; 7g f0; 1; 2; 3; 8g 9000089002 (level 3): 9000080907 (level 2): Students from a class decided to order books for forthcoming 0 holiday. The book-shop in the neighborhood had two Find BA (the complement to B in A) for A = Z and bestsellers on the stock: a crime novel and a horror stories. B = fx 2 Z; jxj > 3g. • There were 31 students in the class in total. • From this total, 22 students bought horror stories. {−3; −2; −1; 0; 1; 2; 3g {−2; −1; 0; 1; 2g • Altogether 12 students bought just one of the books. • Two students did not buy any of these books. f0; 1; 2; 3g f1; 2; 3g How many students bought the crime novel? 24 7 5 9000080908 (level 2): Find the set difference A n B for A = {−2; −1; 0; 1; 2g and B = fx 2 Z; x < 2g. 9000089003 (level 3): Students from a class bought a snack in the school lunch-room. f2g {−2; −1; 0; 1; 2g f0; 1g • There were 31 students in the class in total. • Altogether 8 students had snack from their home and they did not buy anything. ; • Altogether 12 students bought hamburger and 15 students bought hot-dog. How many students bought both hamburger and hot-dog. 9000080909 (level 2): Find the set difference B n A for A = fx 2 ; x < 2g and Z 4 19 8 B = fx 2 Z; x < 5g. f2; 3; 4g fx 2 Z; x < 2g f3; 4g 9000089004 (level 3): Students in a university are allowed to buy either lunch or ; dinner in a student’s canteen. • There are 129 freshmen students in total. • Altogether 116 freshmen students buy the lunch or dinner. 9000080910 (level 2): • From this amount, 62 students buy just one meal. • The number of freshmen students which buy lunch is bigger Find the set difference A n B for A = fx 2 Z; jxj < 3g and by 46 than the number of freshmen students which buy dinner. B = fx 2 N; x < 5g. How many freshmen students buy only the dinner? {−2; −1; 0g ; {−2; −1g f3; 4g 8 54 62 3 9000089005 (level 3): Complete the following statement: „The number is divisible by There are two types of cheese in a shop. Altogether 153 two if and only if ...” customer were in the shop during a day. • From this total, 65 customers bought the first cheese. the last digit of this number is even. • From the same total, 49 customers bought the second cheese. the sum of its digits is divisible by two. • 20% of customers which bought at least one of the types of cheese bought actually both types. the sum of its digits is even. How many customers did not buy any of these two cheeses? the last digit of this number is 2; 3; 6 or 8. 58 39 19 9000089006 (level 3): 9000115602 (level 1): A questionnaire on popular singers has been filled by 200 girls Complete the following statement: „The number is divisible by from a high-school. The girls had to mark the singers from three if and only if ...” three most popular (Justin Timberlake, Justin Bieber and Axl Rose) which they like. the sum of its digits is divisible by three. • Justin Timberlake got 78 votes, Justin Bieber got 75 votes and Axl Rose 101 votes. the number constituted from the last two digits is divisi- • There were 28 girls which voted for all three singers. ble by three. • There were 22 girls which voted for two singers. One half of this amount are fans of the pair Bieber and Rose. the sum of its digits is odd. • The number of the girls which like only Justin Bieber is smaller by 7 comparing to the number of girls which like only the last digit of this number is 3; 6 or 9. Justin Timberlake. How many girls did not like any of these three singers? 9000115603 (level 1): 24 32 11 Complete the following statement: „The number is divisible by four if and only if ...” 9000089007 (level 3): the number constituted from the last two digits is divisi- A class in the school has 35 students.