# Lambda Calculus Practice Problems¶

Evaluate each of these expressions, that is, $$\beta$$-reduce each of the following expressions until you cannot $$\beta$$-reduce any more.

1. $$(λp.λq.pqp)(λx.λy.y)(λx.λy.x)$$
2. $$(λp.λq.ppq)(λx.λy.y)(λx.λy.x)$$
3. $$(λp.p(λx.λy.y)(λx.λy.x))(λx.λy.y)$$
4. $$(λp.λa.λb.pab)(λx.λy.y)(λf.(λx.fx))(λf.(λx.f(fx)))$$
5. $$(λc.c(λx.λy.y))(λx.λy.λf.fxy)(λf.λx.f(fx))(λf.λx.f(f(fx)))$$