Example: Logical Operators

Operators > Boolean Operators > Example: Logical Operators

You can simplify expressions that contain logical operators and comparison operators. Simplification of such expressions returns a refined result that is easier to understand.

Logical AND

• Evaluate an expression with two intervals that do not have any intersections.

• Evaluate the expression with the keyword simplify. Since the intervals do not overlap, the result is 0 (false).

• Evaluate an expression with two intervals that intersect in one point. Evaluating with the keyword simplify returns the intersection point.

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  <math resultRef="10">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <and />
        <apply>
          <greaterOrEqual />
          <apply>
            <minus />
            <apply>
              <nthRoot />
              <placeholder />
              <id xml:space="preserve">a</id>
            </apply>
            <real>2</real>
          </apply>
          <real>0</real>
        </apply>
        <apply>
          <greaterOrEqual />
          <apply>
            <plus />
            <apply>
              <neg />
              <apply>
                <nthRoot />
                <placeholder />
                <id xml:space="preserve">a</id>
              </apply>
            </apply>
            <real>2</real>
          </apply>
          <real>0</real>
        </apply>
      </apply>
      <command>
        <id xml:space="preserve">simplify</id>
      </command>
      <symResult>
        <apply>
          <equal />
          <apply>
            <nthRoot />
            <placeholder />
            <id xml:space="preserve" labels="VARIABLE">a</id>
          </apply>
          <real>2</real>
        </apply>
      </symResult>
    </symEval>
  </math>
</region>

• Evaluate an expression with two intervals where one interval is included in the other. Evaluating with the keyword simplify returns the smallest interval.

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  <math resultRef="11">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <and />
        <apply>
          <greaterOrEqual />
          <apply>
            <minus />
            <apply>
              <nthRoot />
              <placeholder />
              <id xml:space="preserve">a</id>
            </apply>
            <real>2</real>
          </apply>
          <real>0</real>
        </apply>
        <apply>
          <greaterOrEqual />
          <apply>
            <minus />
            <apply>
              <nthRoot />
              <placeholder />
              <id xml:space="preserve">a</id>
            </apply>
            <real>1</real>
          </apply>
          <real>0</real>
        </apply>
      </apply>
      <command>
        <id xml:space="preserve">simplify</id>
      </command>
      <symResult>
        <apply>
          <greaterOrEqual />
          <apply>
            <nthRoot />
            <placeholder />
            <id xml:space="preserve" labels="VARIABLE">a</id>
          </apply>
          <real>2</real>
        </apply>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

• Evaluate an expression with two intervals that intersect in a specific range. Evaluating with the keyword simplify returns the intersection range.

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  <math resultRef="12">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <and />
        <apply>
          <greaterOrEqual />
          <apply>
            <minus />
            <apply>
              <neg />
              <apply>
                <nthRoot />
                <placeholder />
                <id xml:space="preserve">a</id>
              </apply>
            </apply>
            <real>1</real>
          </apply>
          <real>0</real>
        </apply>
        <apply>
          <greaterOrEqual />
          <apply>
            <plus />
            <apply>
              <nthRoot />
              <placeholder />
              <id xml:space="preserve">a</id>
            </apply>
            <real>2</real>
          </apply>
          <real>0</real>
        </apply>
      </apply>
      <command>
        <id xml:space="preserve">simplify</id>
      </command>
      <symResult>
        <apply>
          <lessOrEqual />
          <apply>
            <lessOrEqual />
            <real>-2</real>
            <apply>
              <nthRoot />
              <placeholder />
              <id xml:space="preserve" labels="VARIABLE">a</id>
            </apply>
          </apply>
          <real>-1</real>
        </apply>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

Logical OR

• Evaluate an expression with two intervals that overlap the whole domain. Evaluating with the keyword simplify returns the domain.

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  <math resultRef="13">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <or />
        <apply>
          <greaterOrEqual />
          <apply>
            <nthRoot />
            <placeholder />
            <id xml:space="preserve">a</id>
          </apply>
          <real>2</real>
        </apply>
        <apply>
          <greaterOrEqual />
          <real>2</real>
          <apply>
            <nthRoot />
            <placeholder />
            <id xml:space="preserve">a</id>
          </apply>
        </apply>
      </apply>
      <command>
        <id xml:space="preserve">simplify</id>
      </command>
      <symResult>
        <apply>
          <elementOf />
          <apply>
            <nthRoot />
            <placeholder />
            <id xml:space="preserve" labels="VARIABLE">a</id>
          </apply>
          <id xml:space="preserve" labels="CONSTANT">ℝ</id>
        </apply>
      </symResult>
    </symEval>
  </math>
</region>

• Evaluate an expression where the intervals exclude only one point. Evaluating with the keyword simplify returns the excluded point.

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  <math resultRef="14">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <or />
        <apply>
          <lessThan />
          <apply>
            <nthRoot />
            <placeholder />
            <id xml:space="preserve">a</id>
          </apply>
          <real>2</real>
        </apply>
        <apply>
          <lessThan />
          <real>2</real>
          <apply>
            <nthRoot />
            <placeholder />
            <id xml:space="preserve">a</id>
          </apply>
        </apply>
      </apply>
      <command>
        <id xml:space="preserve">simplify</id>
      </command>
      <symResult>
        <apply>
          <notEqual />
          <apply>
            <nthRoot />
            <placeholder />
            <id xml:space="preserve" labels="VARIABLE">a</id>
          </apply>
          <real>2</real>
        </apply>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

• Evaluate an expression with two intervals, when one is contained inside the other. Evaluating with the keyword simplify returns the largest interval.

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  <math resultRef="15">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <or />
        <apply>
          <greaterOrEqual />
          <apply>
            <nthRoot />
            <placeholder />
            <id xml:space="preserve">a</id>
          </apply>
          <real>2</real>
        </apply>
        <apply>
          <greaterThan />
          <apply>
            <nthRoot />
            <placeholder />
            <id xml:space="preserve">a</id>
          </apply>
          <real>-2</real>
        </apply>
      </apply>
      <command>
        <id xml:space="preserve">simplify</id>
      </command>
      <symResult>
        <apply>
          <greaterThan />
          <apply>
            <nthRoot />
            <placeholder />
            <id xml:space="preserve" labels="VARIABLE">a</id>
          </apply>
          <real>-2</real>
        </apply>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>