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3. Write the following statement in symbolic language as a direct implication and de- termine its contrapositive form. (a) If y is an integer, then every integer of the form 4y + 2 is even. (b) If y is an integer and x is an even integer then xy is an even integer.

3. Write the following statement in symbolic language as a direct implication and de- termine its contrapositive form. (a) If y is an integer, then every integer of the form 4y + 2 is even. (b) If y is an integer and x is an even integer then xy is an even integer.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Problem Statement:** Write the following statement in symbolic language as a direct implication and determine its contrapositive form. (a) If \( y \) is an integer, then every integer of the form \( 4y + 2 \) is even. (b) If \( y \) is an integer and \( x \) is an even integer, then \( xy \) is an even integer. **Explanation:** - For part (a), the statement can be expressed in symbolic form using logic notation. If \( y \) is an integer, this implies \( Q \), where \( Q \) is the condition that \( 4y + 2 \) is even. - **Symbolic Form:** \( P \implies Q \) where \( P \) is the condition that \( y \) is an integer. - **Contrapositive:** The contrapositive of a statement says that if the conclusion is false, then the premise must also be false. Hence, the contrapositive form is: If \( 4y + 2 \) is not even, then \( y \) is not an integer. - For part (b), the statement involves both \( y \) and \( x \). - **Symbolic Form:** \( (P \land R) \implies S \) where \( P \) is that \( y \) is an integer, \( R \) is that \( x \) is an even integer, and \( S \) is that \( xy \) is an even integer. - **Contrapositive:** If \( xy \) is not an even integer, then either \( y \) is not an integer or \( x \) is not an even integer.
Transcribed Image Text:**Problem Statement:** Write the following statement in symbolic language as a direct implication and determine its contrapositive form. (a) If \( y \) is an integer, then every integer of the form \( 4y + 2 \) is even. (b) If \( y \) is an integer and \( x \) is an even integer, then \( xy \) is an even integer. **Explanation:** - For part (a), the statement can be expressed in symbolic form using logic notation. If \( y \) is an integer, this implies \( Q \), where \( Q \) is the condition that \( 4y + 2 \) is even. - **Symbolic Form:** \( P \implies Q \) where \( P \) is the condition that \( y \) is an integer. - **Contrapositive:** The contrapositive of a statement says that if the conclusion is false, then the premise must also be false. Hence, the contrapositive form is: If \( 4y + 2 \) is not even, then \( y \) is not an integer. - For part (b), the statement involves both \( y \) and \( x \). - **Symbolic Form:** \( (P \land R) \implies S \) where \( P \) is that \( y \) is an integer, \( R \) is that \( x \) is an even integer, and \( S \) is that \( xy \) is an even integer. - **Contrapositive:** If \( xy \) is not an even integer, then either \( y \) is not an integer or \( x \) is not an even integer.
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Step 1

a.if ∃y∈Z→∀x∈Z,x=4y+2∈2Z

b.if ∃y∈Z∨x∈Z→XY∈Z

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