Vidaara.orgClass 11 · Mathematics
CodeVID-M11-16-MR-02
Quantifiers & Implications — Assignment
Name: ____________________
Roll No.: __________
Date: ____________
General Instructions
- All questions are compulsory.
- Section A carries 1 mark each, Section B 2 marks, Section C 3 marks and Section D 5 marks.
- Show all working for Sections B, C and D. Only final answers are given at the end — for full solutions, raise your doubts with your teacher.
Section A — Multiple Choice Questions
5 × 1 = 5 marks
1.
The symbol $\forall$ means:
- A.there exists
- B.for all
- C.implies
- D.not
2.
The negation of "$\exists x : P(x)$" is:
- A.$\exists x : \sim P(x)$
- B.$\forall x,\ \sim P(x)$
- C.$\forall x,\ P(x)$
- D.$\sim \exists x$
3.
In $p \Rightarrow q$, $q$ is:
- A.sufficient for $p$
- B.necessary for $p$
- C.equivalent to $p$
- D.the negation of $p$
4.
The converse of $p \Rightarrow q$ is:
- A.$\sim q \Rightarrow \sim p$
- B.$q \Rightarrow p$
- C.$\sim p \Rightarrow \sim q$
- D.$p \Leftrightarrow q$
5.
Which is logically equivalent to $p \Rightarrow q$?
- A.its converse
- B.its contrapositive
- C.$p \wedge q$
- D.$p \vee q$
Section B — Short Answer (2 marks)
4 × 2 = 8 marks
6.
Negate: "Every real number is positive."
7.
Write the contrapositive of "If it rains, the match is cancelled".
8.
Rewrite "$p$ only if $q$" as an if-then statement.
9.
State the truth value of "If $3 > 5$, then $2 + 2 = 4$".
Section C — Short Answer (3 marks)
4 × 3 = 12 marks
10.
For "If a number is a multiple of $9$, then it is a multiple of $3$", write the converse and say whether it is true.
11.
Write the contrapositive of "If $x^2$ is odd, then $x$ is odd".
12.
Express "$x = 2$ if and only if $x^2 = 4$ and $x > 0$" as two conditionals.
13.
Negate "For all integers $n$, $n^2 \ge n$".
Section D — Long Answer (5 marks)
2 × 5 = 10 marks
14.
For "If a triangle is equilateral, then it is isosceles": write (a) the converse, (b) the contrapositive, and state the truth value of each (original is true).
15.
Construct the truth table of $p \Rightarrow q$ and verify it agrees with its contrapositive $\sim q \Rightarrow \sim p$ in all four rows.
Answer Key
Section A — Multiple Choice Questions
- (B) for all
- (B) $\forall x,\ \sim P(x)$
- (B) necessary for $p$
- (B) $q \Rightarrow p$
- (B) its contrapositive
Section B — Short Answer (2 marks)
- "There exists a real number that is not positive."
- "If the match is not cancelled, then it did not rain."
- "If $p$, then $q$" ($p \Rightarrow q$).
- True — the hypothesis is false, so the conditional is vacuously true.
Section C — Short Answer (3 marks)
- Converse: "If a number is a multiple of $3$, then it is a multiple of $9$" — false ($6$ is a counter-example).
- "If $x$ is even, then $x^2$ is even."
- "If $x = 2$ then $x^2 = 4$ and $x > 0$" and "If $x^2 = 4$ and $x > 0$ then $x = 2$".
- "There exists an integer $n$ such that $n^2 < n$."
Section D — Long Answer (5 marks)
- (a) Converse: "If a triangle is isosceles, then it is equilateral" — false. (b) Contrapositive: "If a triangle is not isosceles, then it is not equilateral" — true.
- Both columns read T, F, T, T for $(p,q)=$ (T,T),(T,F),(F,T),(F,F), so the two are equivalent.
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