Logical implication symbols (⇒, ⇔, ⇐)

9 Explain why each of the following statements is false. State in each case which of the symbols ⇒, ⟸ or ⇔ would make the statement true.

1. ABCD is a square \(\Leftrightarrow\) the diagonals of quadrilateral ABCD intersect at \(90 ^ { \circ }\)
2. \(x ^ { 2 }\) is an integer \(\Rightarrow x\) is an integer

1

1. Statement P is \(a + b = 4\).
   Statement Q is \(\quad a = 1\) and \(b = 3\).
   Which one of the following is correct? $$\mathrm { P } \Rightarrow \mathrm { Q } , \quad \mathrm { P } \Leftrightarrow \mathrm { Q } , \quad \mathrm { P } \Leftarrow \mathrm { Q }$$
2. Statement R is \(\quad x = 2\). Statement S is \(\quad x ^ { 2 } = 4\). Which one of the following is correct? $$R \Rightarrow S , \quad R \Leftrightarrow S , \quad R \Leftarrow S$$

3 Select the best statement from $$\begin{aligned} & \mathrm { P } \Rightarrow \mathrm { Q } \\ & \mathrm { P } \Leftarrow \mathrm { Q } \\ & \mathrm { P } \Leftrightarrow \mathrm { Q } \end{aligned}$$ none of the above
to describe the relationship between P and Q in each of the following cases.

1. P: WXYZ is a quadrilateral with 4 equal sides
   \(\mathrm { Q } : \mathrm { WXYZ }\) is a square
2. P: \(n\) is an odd integer Q : \(\quad ( n + 1 ) ^ { 2 }\) is an odd integer
3. P : \(n\) is greater than 1 and \(n\) is a prime number Q : \(\sqrt { n }\) is not an integer

7 In each of the following cases choose one of the statements $$\mathrm { P } \Rightarrow \mathrm { Q } \quad \mathrm { P } \Leftrightarrow \mathrm { Q } \quad \mathrm { P } \Leftarrow \mathrm { Q }$$ to describe the complete relationship between P and Q .

1. P: \(x ^ { 2 } + x - 2 = 0\) Q: \(x = 1\)
2. P: \(y ^ { 3 } > 1\) Q: \(y > 1\)

10 Select the best statement from $$\begin{aligned} & \mathbf { P } \Rightarrow \mathbf { Q } \\ & \mathbf { P } \Leftarrow \mathbf { Q } \\ & \mathbf { P } \Leftrightarrow \mathbf { Q } \end{aligned}$$ none of the above
to describe the relationship between P and Q in each of the following cases.

1. P : WXYZ is a quadrilateral with 4 equal sides
   \(\mathrm { Q } : \mathrm { WXYZ }\) is a square
2. P: \(n\) is an odd integer Q : \(\quad ( n + 1 ) ^ { 2 }\) is an odd integer
3. P: \(n\) is greater than 1 and \(n\) is a prime number Q : \(\sqrt { n }\) is not an integer

9 Complete each of the following by putting the best connecting symbol ( \(\Leftrightarrow , \Leftarrow\) or ⇒ ) in the box. Explain your choice, giving full reasons.

1. \(n ^ { 3 } + 1\) is an odd integer □ \(n\) is an even integer
2. \(( x - 3 ) ( x - 2 ) > 0\) □ \(x > 3\) Section B (36 marks)

3 In each case, choose one of the statements $$\mathbf { P } \Rightarrow \mathbf { Q } \quad \mathbf { P } \Leftarrow \mathbf { Q } \quad \mathbf { P } \Leftrightarrow \mathbf { Q }$$ to describe the complete relationship between P and Q .

1. For \(n\) an integer: P: \(n\) is an even number
   Q: \(n\) is a multiple of 4
2. For triangle ABC : P: \(\quad \mathrm { B }\) is a right-angle
   Q: \(\quad \mathrm { AB } ^ { 2 } + \mathrm { BC } ^ { 2 } = \mathrm { AC } ^ { 2 }\)

3 In each of the following cases choose one of the statements $$P \Rightarrow Q \quad P \Leftarrow Q \quad P \Leftrightarrow Q$$ to describe the relationship between \(P\) and \(Q\).

1. \(P : y = 3 x ^ { 5 } - 4 x ^ { 2 } + 12 x\)
   \(Q : \frac { \mathrm { d } y } { \mathrm {~d} x } = 15 x ^ { 4 } - 8 x + 12\)
2. \(\quad P : x ^ { 5 } - 32 = 0\) where \(x\) is real
   \(Q : x = 2\)
3. \(\quad P : \ln y < 0\)
   \(Q : y < 1\)

4 In each of the following cases choose one of the statements $$\mathrm { P } \Rightarrow \mathrm { Q } \quad \mathrm { P } \Leftrightarrow \mathrm { Q } \quad \mathrm { P } \Leftarrow \mathrm { Q }$$ to describe the complete relationship between P and Q .

1. P: \(x ^ { 2 } + x - 2 = 0\) Q: \(x = 1\)
2. \(\mathrm { P } : y ^ { 3 } > 1\) Q: \(y > 1\)

1 Which of these statements is correct? Tick one box. $$\begin{aligned} & x = 2 \Rightarrow x ^ { 2 } = 4 \\ & x ^ { 2 } = 4 \Rightarrow x = 2 \\ & x ^ { 2 } = 4 \Leftrightarrow x = 2 \\ & x ^ { 2 } = 4 \Rightarrow x = - 2 \end{aligned}$$