Counterexamples, "not", "and", "or", and Venn diagrams

Grade 10 Math Worksheet - Challenge Me

Circle the best answer.

1. 1.

   Proof by contradiction starts by assuming the opposite of what is to be shown, then reaches a contradiction. Esme proves 'a triangle cannot have two right angles' by contradiction. Which assumption should the proof start from?

   • Assume the angle sum is 180°, then show a triangle has two right angles
   • Assume a triangle has two right angles, then show the sum exceeds 180°
   • Assume a triangle has no right angle, then show the angle sum is not 180°, a contradiction
   • Assume a triangle has three right angles, then look for a contradiction
2. 2.

   A counterexample has perimeter 24 and area greater than 35. Ari claims: if a rectangle has perimeter 24, its area is at most 35. Which rectangle is a counterexample?

   • A 6 by 6 square, with area 36
   • A 6 by 7 rectangle, with area 42
   • A 5 by 7 rectangle, with area 35
   • A 4 by 8 rectangle, with area 32
3. 3.

   Of Bo's 60 classmates, 35 take geometry, 28 take art and 12 take neither. How many take both?

   • 3
   • 15
   • 48
   • 20

4. Read this first

   Use this

   Art 20, Band 18, Chess 15. Art and Band 7, Art and Chess 6, Band and Chess 5, each including the students who like all three. All three: 3.

5. 4.

   Jun surveys students about Art, Band and Chess. Using the counts on the card, how many like at least one?

   • 38
   • 53
   • 32
   • 35
6. 5.

   S is true. Two statements are equivalent when they are true or false together. S: if a quadrilateral has four equal sides, then its diagonals are perpendicular. Milo wants a true statement equivalent to S. Which one?

   • If a quadrilateral's diagonals are not perpendicular, then its four sides are not all equal
   • If a quadrilateral's diagonals are perpendicular, then it has four equal sides
   • If a quadrilateral does not have four equal sides, then its diagonals are not perpendicular
   • A quadrilateral has four equal sides and its diagonals are not perpendicular

7. Use this

   Total 40. Art 17, Band 15, Chess 12. Art and Band 6, Art and Chess 4, Band and Chess 5, each including those in all three clubs. 9 are in no club. The number in all three clubs is not given.

8. 6.

   Kira's class has 40 students. Using the counts on the card, how many are in all three clubs?

   • 13
   • 9
   • 11
   • 2
9. 7.

   A counterexample makes xy = 0 true but x ≠ 0. Tala claims: if xy = 0, then x = 0. Which pair of values is a counterexample?

   • x = 0 and y = 5
   • x = 0 and y = 0
   • x = 2 and y = 3
   • x = 5 and y = 0

10. Use this

    The contrapositive of 'if P then Q' is 'if not Q then not P'. P: n² is even. Q: n is even.

11. 8.

    Noah proves 'if n² is even, then n is even' by contrapositive. Which should the proof start from?

    • Assume n² is odd, then show n is odd
    • Assume n is odd, then show n² is odd
    • Assume n is even, then show n² is even
    • Assume n² is even, then show n is even
12. 9.

    For every triangle, at least one side is shorter than 10. Ren says: for every triangle there is a side shorter than 10. Which statement is its negation?

    • There is a triangle with a side longer than 10
    • For every triangle there is a side longer than 10
    • There is a triangle whose sides are all at least 10
    • For every triangle, all sides are at least 10
13. 10.

    A new club survey has Chess only 0, Art+Chess only 3, Band+Chess only 2, all three 1. Pia claims everyone in Chess is also in Art or Band. Is it true?

    • True: the Chess-only region is empty, so every Chess member is in Art or Band
    • True: Chess has fewer members than Art, so they must sit inside Art's circle
    • False: Chess has 6 members, and only the 3 in Art and Chess count as covered
    • False: the 9 students in no club break the claim about Chess members