TMUA: Quantifiers and negation

Paper 2: quantifiers and negation.

Practise quantifiers and negation →Timed TMUA paper

What the specification covers

Key ideas

Common mistakes

Exam tip

Negate from the outside in: flip each quantifier, then negate the inside.

Worked example

Worked example

Which of the following is the negation of the statement: 'For every real number \(x\) there is an integer \(n\) with \(n>x\)'?

  1. There is a real number \(x\) and an integer \(n\) with \(n\le x\).
  2. For every real number \(x\), every integer \(n\) satisfies \(n\le x\).
  3. For every real number \(x\) there is an integer \(n\) with \(n\le x\).
  4. There is a real number \(x\) such that every integer \(n\) satisfies \(n\le x\).
  5. There is an integer \(n\) such that every real number \(x\) satisfies \(n\le x\).

Answer: D: There is a real number \(x\) such that every integer \(n\) satisfies \(n\le x\).

'Not (for every \(x\), P(\(x\)))' is 'there is an \(x\) with not P(\(x\))'. And 'not (there is an \(n\) with \(n>x\))' is 'every \(n\) has \(n\le x\)'. So the negation is 'there is a real \(x\) such that every integer \(n\) satisfies \(n\le x\)'. (The original statement is true, so its negation is false.)

Practice questions

Try each question before opening the solution. No calculator: the TMUA does not allow one. Every answer here was re-checked independently by computer before it was published.

Question 1

Which of the following is the negation of the statement '\(x>2\) and \(y\le5\)'?

  1. \(x\le2\) or \(y\ge5\)
  2. \(x>2\) or \(y\le5\)
  3. \(x<2\) or \(y>5\)
  4. \(x\le2\) or \(y>5\)
  5. \(x\le2\) and \(y>5\)
Show the answer and solution

Answer: D: \(x\le2\) or \(y>5\)

'Not (A and B)' is '(not A) or (not B)'. Not \(x>2\) is \(x\le2\); not \(y\le5\) is \(y>5\).

Question 2

Let \(S=\{1,2,3,4,5,6\}\). Which of the following statements are true?
I   For every \(a\) in \(S\) there is a \(b\) in \(S\) with \(a+b=7\).
II   There is a \(b\) in \(S\) such that for every \(a\) in \(S\), \(ab\) is even.
III   For every \(a\) in \(S\) there is a \(b\) in \(S\), with \(b\ne a\), such that \(a\) divides \(b\).

  1. I and III only
  2. III only
  3. I only
  4. I and II only
  5. none of them
  6. II only
  7. II and III only
  8. I, II and III
Show the answer and solution

Answer: D: I and II only

I: take \(b=7-a\), which is in \(S\). II: \(b=2\) works for every \(a\). III fails for \(a=4\): the only multiple of 4 in \(S\) is 4 itself.

Question 3

Which of the following is the negation of the statement '\(x\ge1\) or \(y\ge0\)'?

  1. \(x<1\) or \(y<0\)
  2. \(x<1\) and \(y<0\)
  3. \(x\ge1\) and \(y\ge0\)
  4. \(x\ge1\) and \(y<0\)
  5. \(x<1\) and \(y\ge0\)
Show the answer and solution

Answer: B: \(x<1\) and \(y<0\)

'Not (P or Q)' is '(not P) and (not Q)'. Not \(x\ge1\) is \(x<1\) and not \(y\ge0\) is \(y<0\).

Question 4

Which of the following is the negation of the statement '\(x\le0\) and \(y\ge-3\)'?

  1. \(x\le0\) or \(y\ge-3\)
  2. \(x>0\) and \(y<-3\)
  3. \(x\le0\) or \(y<-3\)
  4. \(x>0\) or \(y<-3\)
  5. \(x>0\) or \(y\ge-3\)
Show the answer and solution

Answer: D: \(x>0\) or \(y<-3\)

'Not (P and Q)' is '(not P) or (not Q)'. Not \(x\le0\) is \(x>0\) and not \(y\ge-3\) is \(y<-3\).

Question 5

Which of the following is the negation of the statement '\(x<0\) and \(y<6\)'?

  1. \(x\ge0\) or \(y\ge6\)
  2. \(x<0\) or \(y\ge6\)
  3. \(x\ge0\) and \(y\ge6\)
  4. \(x\ge0\) or \(y<6\)
  5. \(x<0\) or \(y<6\)
Show the answer and solution

Answer: A: \(x\ge0\) or \(y\ge6\)

'Not (P and Q)' is '(not P) or (not Q)'. Not \(x<0\) is \(x\ge0\) and not \(y<6\) is \(y\ge6\).

Question 6

Which of the following is the negation of the statement '\(x<2\) and \(y<-3\)'?

  1. \(x\ge2\) and \(y\ge-3\)
  2. \(x<2\) or \(y\ge-3\)
  3. \(x<2\) or \(y<-3\)
  4. \(x\ge2\) or \(y<-3\)
  5. \(x\ge2\) or \(y\ge-3\)
Show the answer and solution

Answer: E: \(x\ge2\) or \(y\ge-3\)

'Not (P and Q)' is '(not P) or (not Q)'. Not \(x<2\) is \(x\ge2\) and not \(y<-3\) is \(y\ge-3\).

Question 7

Which of the following is the negation of the statement '\(x\le0\) and \(y\ge5\)'?

  1. \(x\le0\) or \(y<5\)
  2. \(x>0\) or \(y<5\)
  3. \(x>0\) and \(y<5\)
  4. \(x\le0\) or \(y\ge5\)
  5. \(x>0\) or \(y\ge5\)
Show the answer and solution

Answer: B: \(x>0\) or \(y<5\)

'Not (P and Q)' is '(not P) or (not Q)'. Not \(x\le0\) is \(x>0\) and not \(y\ge5\) is \(y<5\).

Question 8

Which of the following is the negation of the statement '\(x\le2\) and \(y<3\)'?

  1. \(x>2\) or \(y<3\)
  2. \(x>2\) or \(y\ge3\)
  3. \(x\le2\) or \(y<3\)
  4. \(x>2\) and \(y\ge3\)
  5. \(x\le2\) or \(y\ge3\)
Show the answer and solution

Answer: B: \(x>2\) or \(y\ge3\)

'Not (P and Q)' is '(not P) or (not Q)'. Not \(x\le2\) is \(x>2\) and not \(y<3\) is \(y\ge3\).

Question 9

Which of the following is the negation of the statement '\(x>-3\) and \(y\ge1\)'?

  1. \(x\le-3\) or \(y<1\)
  2. \(x>-3\) or \(y<1\)
  3. \(x\le-3\) or \(y\ge1\)
  4. \(x>-3\) or \(y\ge1\)
  5. \(x\le-3\) and \(y<1\)
Show the answer and solution

Answer: A: \(x\le-3\) or \(y<1\)

'Not (P and Q)' is '(not P) or (not Q)'. Not \(x>-3\) is \(x\le-3\) and not \(y\ge1\) is \(y<1\).

Question 10

Which of the following is the negation of the statement '\(x\ge6\) and \(y>1\)'?

  1. \(x\ge6\) or \(y>1\)
  2. \(x\ge6\) or \(y\le1\)
  3. \(x<6\) and \(y\le1\)
  4. \(x<6\) or \(y\le1\)
  5. \(x<6\) or \(y>1\)
Show the answer and solution

Answer: D: \(x<6\) or \(y\le1\)

'Not (P and Q)' is '(not P) or (not Q)'. Not \(x\ge6\) is \(x<6\) and not \(y>1\) is \(y\le1\).

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